{ "schema_version": 1, "generated_from": { "claudiverse_commit": "4a88328", "generated_at": "2026-10-10T11:13:12Z" }, "node_types": [ "book", "person", "chapter", "exercise_set", "problem", "problem_form", "problem_shape", "concept", "method", "theorem", "law", "quantity", "unit", "instrument", "experiment", "excerpt", "equation", "capability" ], "edge_types": [ "written_by", "part_of", "taught_in", "practices", "quoted_from", "explains", "appears_in", "states", "relates", "instance_of", "needs", "prerequisite_of", "special_case_of", "generalizes", "uses", "inverse_of", "contrasts_with", "measures", "unit_of", "named_after", "discovered_by", "related_to" ], "book": { "id": "wentworth-plane-geometry-1899", "title": "Plane Geometry", "authors": [ "George Albert Wentworth" ], "year": 1899, "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "transcription": { "source": "Project Gutenberg eBook #33063", "url": "https://www.gutenberg.org/ebooks/33063", "released": "July 3, 2010", "licence_terms": "This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org" }, "file": "books/wentworth-plane-geometry-1899.json" }, "chapters": [ { "id": "wentworth-plane-geometry-1899/ch-introduction", "number": "INTRODUCTION", "title": "INTRODUCTION", "name": "Wentworth 1899, INTRODUCTION", "pages": [ "010", "016" ], "concepts": [ "concept/axiom", "concept/conclusion", "concept/construction", "concept/contradictory-of-a-theorem", "concept/converse-of-a-theorem", "concept/corollary", "concept/dimension", "concept/geometrical-figure", "concept/geometry", "concept/hypothesis", "concept/line", "concept/opposite-of-a-theorem", "concept/plane", "concept/plane-geometry", "concept/point", "concept/postulate", "concept/problem", "concept/proof", "concept/proposition", "concept/scholium", "concept/solid-geometry", "concept/surface", "concept/theorem", "concept/three-dimensional-figure", "method/solution-of-a-problem" ], "excerpts": [ "wentworth-plane-geometry-1899/x-3481c15b88", "wentworth-plane-geometry-1899/x-4e1fd0630d", "wentworth-plane-geometry-1899/x-c6692155d6", "wentworth-plane-geometry-1899/x-ae86efbc5d", "wentworth-plane-geometry-1899/x-e6d26d1d71", "wentworth-plane-geometry-1899/x-7b8052a143" ], "equations": [], "exercise_sets": [] }, { "id": "wentworth-plane-geometry-1899/ch-i", "number": "I", "title": "RECTILINEAR FIGURES", "name": "Wentworth 1899, ch. 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II: THE CIRCLE", "pages": [ "138", "144" ], "concepts": [ "concept/altitude-of-a-triangle", "concept/ambiguous-case", "concept/angle-bisector", "concept/approximation", "concept/arc-of-a-circle", "concept/base-of-a-triangle", "concept/central-angle", "concept/centre-of-a-circle", "concept/chord", "concept/circle", "concept/circular-sector", "concept/circular-segment", "concept/circum-centre", "concept/circumscribed-circle", "concept/commensurable-magnitudes", "concept/concentric-circles", "concept/conjugate-arcs", "concept/constant", "concept/construction", "concept/converse-of-a-theorem", "concept/diagonal", "concept/diameter", "concept/equilateral-triangle", "concept/escribed-circle", "concept/hypotenuse", "concept/in-centre", "concept/incommensurable-magnitudes", "concept/incommensurable-ratio", "concept/inscribed-angle", "concept/inscribed-circle", "concept/isosceles-trapezoid", "concept/isosceles-triangle", "concept/leg-of-a-right-triangle", "concept/limit", 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SIMILAR POLYGONS", "name": "Wentworth 1899, ch. III: PROPORTION\\@. 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V: REGULAR POLYGONS AND CIRCLES", "pages": [ "243", "259" ], "concepts": [ "concept/angle-at-the-centre-of-a-regular-polygon", "concept/apothem", "concept/centre-of-a-regular-polygon", "concept/chord", "concept/circular-sector", "concept/circumscribed-circle", "concept/decagon", "concept/equilateral-polygon", "concept/equilateral-triangle", "concept/extreme-and-mean-ratio", "concept/hexagon", "concept/incommensurable-magnitudes", "concept/inscribed-circle", "concept/isoperimetric-polygon", "concept/limit", "concept/maximum", "concept/minimum", "concept/mutually-equiangular-polygons", "concept/octagon", "concept/pentadecagon", "concept/pentagon", "concept/pi", "concept/radius-of-a-regular-polygon", "concept/regular-polygon", "concept/square", "quantity/circumference", "theorem/area-of-a-circle", "theorem/area-of-a-regular-polygon", "theorem/sum-of-the-interior-angles-of-a-polygon" ], "excerpts": [ "wentworth-plane-geometry-1899/x-9b3ddfe50a", "wentworth-plane-geometry-1899/x-7525f46e3b", "wentworth-plane-geometry-1899/x-7c7dd49317", "wentworth-plane-geometry-1899/x-1224ccd7bd", "wentworth-plane-geometry-1899/x-8f37984bcb", "wentworth-plane-geometry-1899/x-d7bb5d452f", "wentworth-plane-geometry-1899/x-289cb9038e", "wentworth-plane-geometry-1899/x-d0aa9ea5a4", "wentworth-plane-geometry-1899/x-6754238415", "wentworth-plane-geometry-1899/x-961e532717", "wentworth-plane-geometry-1899/x-7e930eced8", "wentworth-plane-geometry-1899/x-bdd2c0f44e" ], "equations": [ "wentworth-plane-geometry-1899/eq-893cd7fb48", "wentworth-plane-geometry-1899/eq-4f814b5170", "wentworth-plane-geometry-1899/eq-fca07b2e3a", "wentworth-plane-geometry-1899/eq-292f074c4a", "wentworth-plane-geometry-1899/eq-fe10d94775", "wentworth-plane-geometry-1899/eq-29776951fe", "wentworth-plane-geometry-1899/eq-2ac9524257", "wentworth-plane-geometry-1899/eq-c03156d208", "wentworth-plane-geometry-1899/eq-56a60be0c2", "wentworth-plane-geometry-1899/eq-0e95c840cf", "wentworth-plane-geometry-1899/eq-ab83981a82", "wentworth-plane-geometry-1899/eq-3ec41d5145", "wentworth-plane-geometry-1899/eq-cef8e23a78", "wentworth-plane-geometry-1899/eq-bb8d819cb3", "wentworth-plane-geometry-1899/eq-7df4945cbf", "wentworth-plane-geometry-1899/eq-8dd5758295", "wentworth-plane-geometry-1899/eq-19b4e5b2d1", "wentworth-plane-geometry-1899/eq-f9f6dce470", "wentworth-plane-geometry-1899/eq-90091e8382", "wentworth-plane-geometry-1899/eq-e39fd6b4df", "wentworth-plane-geometry-1899/eq-989b8e4898", "wentworth-plane-geometry-1899/eq-a7b5b0e974", "wentworth-plane-geometry-1899/eq-4e52308087" ], "exercise_sets": [ "wentworth-plane-geometry-1899/ex-v-1", "wentworth-plane-geometry-1899/ex-misc" ] } ], "excerpts": [ { "id": "wentworth-plane-geometry-1899/x-3481c15b88", "chapter": "wentworth-plane-geometry-1899/ch-introduction", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 10", "location": "INTRODUCTION", "latex": "the straight edge in every part will touch the surface, the faces are called \\textbf{plane surfaces}, or \\indexbf{planes}.", "markdown": "the straight edge in every part will touch the surface, the faces are called **plane surfaces**, or **planes**.", "why": "It gives the learner a physical test for a plane: a straight edge must touch the surface everywhere.", "use": [ "lesson", "website" ], "concepts": [ "concept/plane", "concept/surface" ] }, { "id": "wentworth-plane-geometry-1899/x-4e1fd0630d", "chapter": "wentworth-plane-geometry-1899/ch-introduction", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 11", "location": "INTRODUCTION", "latex": "\\textit{A surface has only two dimensions, length and breadth.}", "markdown": "*A surface has only two dimensions, length and breadth.*", "why": "It shows at once how the dimension count drops from solid to surface to line to point.", "use": [ "lesson" ], "concepts": [ "concept/dimension", "concept/surface" ] }, { "id": "wentworth-plane-geometry-1899/x-c6692155d6", "chapter": "wentworth-plane-geometry-1899/ch-introduction", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 11", "location": "INTRODUCTION", "latex": "\\textit{A point has no dimension, but denotes position simply.}", "markdown": "*A point has no dimension, but denotes position simply.*", "why": "It states plainly that a point marks position and has no size, a useful anchor for beginners.", "use": [ "lesson", "website" ], "concepts": [ "concept/point" ] }, { "id": "wentworth-plane-geometry-1899/x-ae86efbc5d", "chapter": "wentworth-plane-geometry-1899/ch-introduction", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 11", "location": "INTRODUCTION", "latex": "It must be distinctly understood at the outset that the points, lines, surfaces, and solids of Geometry are \\textit{purely ideal}, though they are represented to the eye in a material way.", "markdown": "It must be distinctly understood at the outset that the points, lines, surfaces, and solids of Geometry are *purely ideal*, though they are represented to the eye in a material way.", "why": "It reminds learners that a drawn line is only a picture of a true line, which they must imagine without width.", "use": [ "lesson", "website" ], "concepts": [ "concept/line", "concept/point", "concept/surface", "concept/three-dimensional-figure" ] }, { "id": "wentworth-plane-geometry-1899/x-e6d26d1d71", "chapter": "wentworth-plane-geometry-1899/ch-introduction", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 12", "location": "INTRODUCTION", "latex": "If a surface moves in space, it generates, in general, a solid.", "markdown": "If a surface moves in space, it generates, in general, a solid.", "why": "It gives an intuitive picture of how a solid can be built up from a moving surface.", "use": [ "lesson", "website" ], "concepts": [ "concept/surface", "concept/three-dimensional-figure" ] }, { "id": "wentworth-plane-geometry-1899/x-7b8052a143", "chapter": "wentworth-plane-geometry-1899/ch-introduction", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 14", "location": "INTRODUCTION", "latex": "Thus, Every horse is a quadruped is true, but the converse, Every quad\\-ru\\-ped is a horse, is not true.", "markdown": "Thus, Every horse is a quadruped is true, but the converse, Every quadruped is a horse, is not true.", "why": "A short example that teaches learners not to assume a statement's converse holds because the statement does.", "use": [ "lesson" ], "concepts": [ "concept/converse-of-a-theorem" ] }, { "id": "wentworth-plane-geometry-1899/x-ba6d055e6c", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 65", "location": "RECTILINEAR FIGURES", "latex": "A \\indexbf{convex polygon} is a polygon of which no side, when produced, will enter the polygon.", "markdown": "A **convex polygon** is a polygon of which no side, when produced, will enter the polygon.", "why": "States the test for convexity in terms a learner can check by extending each side.", "use": [ "lesson" ], "concepts": [ "concept/concave-polygon", "concept/convex-polygon" ] }, { "id": "wentworth-plane-geometry-1899/x-b32e5c0e6f", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 66", "location": "RECTILINEAR FIGURES", "latex": "And, \\emph{except in the case of triangles}, two polygons may be mutually equilateral without being mutually equiangular; as, Figs.~6 and 7.", "markdown": "And, *except in the case of triangles*, two polygons may be mutually equilateral without being mutually equiangular; as, Figs. 6 and 7.", "why": "Warns that equal sides do not force equal angles except for triangles, a common mistake.", "use": [ "lesson" ], "concepts": [ "concept/mutually-equiangular-polygons", "concept/mutually-equilateral-polygons", "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/x-facebe6baf", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 67", "location": "RECTILINEAR FIGURES", "latex": "In general, each angle of an equiangular polygon of $n$ sides is equal to $\\displaystyle \\frac{2(n-2)}{n}$ right angles.", "markdown": "In general, each angle of an equiangular polygon of $n$ sides is equal to $\\displaystyle \\frac{2(n-2)}{n}$ right angles.", "why": "Turns the sum-of-angles result into a usable formula for regular-type polygons.", "use": [ "lesson", "website" ], "concepts": [ "concept/equiangular-polygon", "theorem/sum-of-the-angles-of-a-polygon" ] }, { "id": "wentworth-plane-geometry-1899/x-837d29ebcf", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 69", "location": "RECTILINEAR FIGURES", "latex": "Two points are said to be \\textbf{symmetrical} with respect to a third point, called the \\textbf{centre of symmetry}\\label{centresym}, if this third point bisects the straight line which joins them.", "markdown": "Two points are said to be **symmetrical** with respect to a third point, called the **centre of symmetry**, if this third point bisects the straight line which joins them.", "why": "Gives a precise definition of symmetry about a point that a learner can test by measuring a midpoint.", "use": [ "lesson", "website" ], "concepts": [ "concept/centre-of-symmetry", "concept/symmetry" ] }, { "id": "wentworth-plane-geometry-1899/x-a7d77f19da", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 16", "location": "RECTILINEAR FIGURES", "latex": "A \\indexbf{straight line} is a line such that any part of it, however placed on any other part, will lie wholly in that part if its extremities lie in that part, as~$AB$.", "markdown": "A **straight line** is a line such that any part of it, however placed on any other part, will lie wholly in that part if its extremities lie in that part, as $AB$.", "why": "It gives the definition of a straight line by a test any learner can picture: a sliding copy that always stays inside itself.", "use": [ "lesson" ], "concepts": [ "concept/line" ] }, { "id": "wentworth-plane-geometry-1899/x-621a62826d", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 18", "location": "RECTILINEAR FIGURES", "latex": "The size of an angle depends upon the \\emph{extent of opening} of its sides, and not upon the length of its sides.", "markdown": "The size of an angle depends upon the *extent of opening* of its sides, and not upon the length of its sides.", "why": "It corrects the common mistake of thinking that longer arms make a bigger angle.", "use": [ "lesson" ], "concepts": [ "concept/plane-angle" ] }, { "id": "wentworth-plane-geometry-1899/x-ccc6172099", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 21", "location": "RECTILINEAR FIGURES", "latex": "Suppose the straight line $OC$ (Fig.~15) to move in the plane of the paper from coincidence with $OA$, about the point $O$ as a pivot, to the position $OC$; then the line $OC$ describes or generates \\emph{the angle $AOC$}, and the magnitude of the angle $AOC$ depends upon the \\emph{amount of rotation} of the line from the position $OA$ to the position $OC$.", "markdown": "Suppose the straight line $OC$ (Fig. 15) to move in the plane of the paper from coincidence with $OA$, about the point $O$ as a pivot, to the position $OC$; then the line $OC$ describes or generates *the angle $AOC$*, and the magnitude of the angle $AOC$ depends upon the *amount of rotation* of the line from the position $OA$ to the position $OC$.", "why": "It presents an angle as a rotating line, so learners can see why angles can be bigger than a straight angle.", "use": [ "lesson" ], "concepts": [ "concept/plane-angle" ] }, { "id": "wentworth-plane-geometry-1899/x-38137a24a8", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 22", "location": "RECTILINEAR FIGURES", "latex": "The natural angular unit is one complete revolution. But this unit would require us to express the values of most angles by fractions.", "markdown": "The natural angular unit is one complete revolution. But this unit would require us to express the values of most angles by fractions.", "why": "It explains why the degree, not the full turn, is the practical unit for measuring angles.", "use": [ "lesson", "history" ], "concepts": [ "unit/degree-of-angle" ] }, { "id": "wentworth-plane-geometry-1899/x-61570d908d", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 28", "location": "RECTILINEAR FIGURES", "latex": "Fold over $CFA$, on $CF$ as an axis, until it falls on the plane at the right of $CF$.", "markdown": "Fold over $CFA$, on $CF$ as an axis, until it falls on the plane at the right of $CF$.", "why": "It shows the postulate of superposition at work in a proof, so learners see how a figure is moved to compare it.", "use": [ "lesson" ], "concepts": [ "concept/superposition" ] }, { "id": "wentworth-plane-geometry-1899/x-b90976307c", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 24", "location": "RECTILINEAR FIGURES", "latex": "The beginner must not forget that in Plane Geometry all the points of a figure are in the same plane. Without this restriction in Cor.~2, an indefinite number of perpendiculars can be erected at a given point in a given line.", "markdown": "The beginner must not forget that in Plane Geometry all the points of a figure are in the same plane. Without this restriction in Cor. 2, an indefinite number of perpendiculars can be erected at a given point in a given line.", "why": "It warns that uniqueness of the perpendicular depends on working in a single plane.", "use": [ "lesson" ], "concepts": [ "concept/perpendicular", "concept/two-dimensional-figure" ] }, { "id": "wentworth-plane-geometry-1899/x-236943ac8f", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 30", "location": "RECTILINEAR FIGURES", "latex": "The perpendicular is the shortest line that can be drawn to a straight line from an external point.", "markdown": "The perpendicular is the shortest line that can be drawn to a straight line from an external point.", "why": "It states the key fact behind the definition of the distance from a point to a line.", "use": [ "lesson", "website" ], "concepts": [ "concept/distance-from-a-point-to-a-line", "theorem/perpendicular-is-the-shortest-distance-from-a-point-to-a-line" ] }, { "id": "wentworth-plane-geometry-1899/x-9f85565493", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 39", "location": "RECTILINEAR FIGURES", "latex": "A \\indexbf{triangle} is a portion of a plane bounded by three straight lines; as, $ABC$ (Fig.~1).", "markdown": "A **triangle** is a portion of a plane bounded by three straight lines; as, $ABC$ (Fig. 1).", "why": "It gives the learner the plain definition of a triangle as a region of the plane, not just three segments.", "use": [ "lesson", "website" ], "concepts": [ "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/x-67b0818c7e", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 41", "location": "RECTILINEAR FIGURES", "latex": "The sum of the three angles of a triangle is equal to two right angles.", "markdown": "The sum of the three angles of a triangle is equal to two right angles.", "why": "It states the angle-sum theorem that the rest of the chapter's corollaries depend on.", "use": [ "lesson" ], "concepts": [ "theorem/sum-of-the-angles-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/x-1d80a0e940", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 53", "location": "RECTILINEAR FIGURES", "latex": "All points in a plane that satisfy a single geometrical condition lie, in general, in a line or group of lines; and this line or group of lines is called the \\textbf{locus} of the points that satisfy the given condition.", "markdown": "All points in a plane that satisfy a single geometrical condition lie, in general, in a line or group of lines; and this line or group of lines is called the **locus** of the points that satisfy the given condition.", "why": "It defines a locus in one sentence, which is the central idea of the second section.", "use": [ "lesson" ], "concepts": [ "concept/locus" ] }, { "id": "wentworth-plane-geometry-1899/x-b7dea26f46", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 53", "location": "RECTILINEAR FIGURES", "latex": "The word \\emph{locus} (pronounced lo\\'{ }kus) is a Latin word that signifies \\emph{place}. The plural of locus is loci (pronounced lo\\'{ }si).", "markdown": "The word *locus* (pronounced lo kus) is a Latin word that signifies *place*. The plural of locus is loci (pronounced lo si).", "why": "It gives the Latin origin of the term, which helps a learner remember what a locus is.", "use": [ "history" ], "concepts": [ "concept/locus" ] }, { "id": "wentworth-plane-geometry-1899/x-8442de289e", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 44", "location": "RECTILINEAR FIGURES", "latex": "In §~139 we have given two angles and the included side, in §~143 two sides and the included angle; hence, by interchanging the words \\emph{sides} and \\emph{angles}, either theorem is changed to the other. This is called the \\emph{Principle of Duality}\\label{princduality}, or the \\emph{Principle of Reciprocity}\\label{princreciprocity}. The reciprocal of a theorem is not always true, just as the converse of a theorem is not always true.", "markdown": "In § 139 we have given two angles and the included side, in § 143 two sides and the included angle; hence, by interchanging the words *sides* and *angles*, either theorem is changed to the other. This is called the *Principle of Duality*, or the *Principle of Reciprocity*. The reciprocal of a theorem is not always true, just as the converse of a theorem is not always true.", "why": "It warns the learner that swapping sides and angles in a true theorem gives a true theorem, while the reciprocal of a theorem need not be true.", "use": [ "lesson" ], "concepts": [ "concept/converse-of-a-theorem", "concept/principle-of-duality" ] }, { "id": "wentworth-plane-geometry-1899/x-2e21731e52", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 42", "location": "RECTILINEAR FIGURES", "latex": "The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side.", "markdown": "The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side.", "why": "It states the triangle inequality in both forms, which learners often need to apply when checking whether three lengths can form a triangle.", "use": [ "lesson", "website" ], "concepts": [ "theorem/sum-of-two-sides-of-a-triangle-exceeds-the-third" ] }, { "id": "wentworth-plane-geometry-1899/x-24ebe9ca79", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 65", "location": "RECTILINEAR FIGURES", "latex": "A \\indexbf{polygon} is a portion of a plane bounded by straight lines.", "markdown": "A **polygon** is a portion of a plane bounded by straight lines.", "why": "Gives the learner the basic picture of a polygon as a flat region fenced in by straight sides.", "use": [ "lesson", "website" ], "concepts": [ "concept/polygon" ] }, { "id": "wentworth-plane-geometry-1899/x-c4147afd14", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 65", "location": "RECTILINEAR FIGURES", "latex": "A \\indexbf{concave polygon} is a polygon of which two or more sides, if produced, will enter the polygon.", "markdown": "A **concave polygon** is a polygon of which two or more sides, if produced, will enter the polygon.", "why": "Shows the difference between convex and concave shapes by what happens when the sides are extended.", "use": [ "lesson" ], "concepts": [ "concept/concave-polygon", "concept/convex-polygon" ] }, { "id": "wentworth-plane-geometry-1899/x-28b87bb4f2", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 66", "location": "RECTILINEAR FIGURES", "latex": "Two polygons are \\emph{equal} when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed; for if the polygons are applied to each other, the corresponding triangles will coincide, and hence the polygons will coincide and be equal.", "markdown": "Two polygons are *equal* when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed; for if the polygons are applied to each other, the corresponding triangles will coincide, and hence the polygons will coincide and be equal.", "why": "Explains what it means for two polygons to be equal by reducing the idea to equal triangles that coincide when laid on each other.", "use": [ "lesson" ], "concepts": [ "concept/congruent-figures", "concept/diagonal" ] }, { "id": "wentworth-plane-geometry-1899/x-77bdfd654f", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 73", "location": "RECTILINEAR FIGURES", "latex": "If a known truth \\emph{suggests} the required proof, it is best to use the synthetic form at once. If no proof occurs to the mind, it is necessary to use the analytic method to \\emph{discover} the proof, and then the synthetic proof may be given.", "markdown": "If a known truth *suggests* the required proof, it is best to use the synthetic form at once. If no proof occurs to the mind, it is necessary to use the analytic method to *discover* the proof, and then the synthetic proof may be given.", "why": "Tells the learner when to build a proof forward and when to work backward to find it first.", "use": [ "lesson" ], "concepts": [ "method/analytic-method", "method/synthetic-method" ] }, { "id": "wentworth-plane-geometry-1899/x-06e9780849", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 67", "location": "RECTILINEAR FIGURES", "latex": "The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.", "markdown": "The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.", "why": "Gives the general angle-sum rule that the learner can test on any polygon they draw.", "use": [ "lesson", "website" ], "concepts": [ "theorem/sum-of-the-angles-of-a-polygon" ] }, { "id": "wentworth-plane-geometry-1899/x-061d63fd16", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 69", "location": "RECTILINEAR FIGURES", "latex": "A figure is symmetrical with respect to a point as a centre of symmetry, if the point bisects every straight line drawn through it and terminated by the boundary of the figure.", "markdown": "A figure is symmetrical with respect to a point as a centre of symmetry, if the point bisects every straight line drawn through it and terminated by the boundary of the figure.", "why": "Defines point symmetry in terms the learner can check by drawing chords through a candidate centre.", "use": [ "lesson" ], "concepts": [ "concept/centre-of-symmetry", "concept/symmetry" ] }, { "id": "wentworth-plane-geometry-1899/x-0f94897358", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 82", "location": "RECTILINEAR FIGURES", "latex": "The lines joining the middle points of the sides of a square, taken in order, enclose a square.", "markdown": "The lines joining the middle points of the sides of a square, taken in order, enclose a square.", "why": "A surprising result that rewards the learner who draws it and checks it, a good hook for a website.", "use": [ "website" ], "concepts": [ "concept/square" ] }, { "id": "wentworth-plane-geometry-1899/x-6404f08d5c", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 109", "location": "THE CIRCLE", "latex": "In the same circle or in equal circles, two central angles have the same ratio as their intercepted arcs.", "markdown": "In the same circle or in equal circles, two central angles have the same ratio as their intercepted arcs.", "why": "States the chapter's central result in one sentence, so a learner knows what the proof that follows establishes.", "use": [ "lesson" ], "concepts": [ "concept/arc-of-a-circle", "concept/central-angle", "theorem/central-angles-have-the-same-ratio-as-their-intercepted-arcs" ] }, { "id": "wentworth-plane-geometry-1899/x-61192f61ef", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 111", "location": "THE CIRCLE", "latex": "An inscribed angle is measured by half the arc intercepted between its sides.", "markdown": "An inscribed angle is measured by half the arc intercepted between its sides.", "why": "Gives the inscribed-angle rule a learner can apply directly to find an angle from an arc.", "use": [ "lesson" ], "concepts": [ "concept/inscribed-angle", "theorem/inscribed-angle-is-measured-by-half-its-intercepted-arc" ] }, { "id": "wentworth-plane-geometry-1899/x-6d8bd6d617", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 111", "location": "THE CIRCLE", "latex": "A circumference is divided into $360$~equal parts, called \\emph{degrees}; and therefore a unit angle at the centre intercepts a unit arc on the circumference.", "markdown": "A circumference is divided into $360$ equal parts, called *degrees*; and therefore a unit angle at the centre intercepts a unit arc on the circumference.", "why": "It explains why angle and arc can share one numerical measure, which is the basis for measuring angles by arcs.", "use": [ "lesson" ], "concepts": [ "concept/central-angle", "concept/plane-angle", "theorem/central-angle-is-measured-by-its-intercepted-arc", "unit/degree-of-angle" ] }, { "id": "wentworth-plane-geometry-1899/x-453bf214bf", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 113", "location": "THE CIRCLE", "latex": "An angle formed by two chords intersecting within the circumference is measured by half the sum of the intercepted arcs.", "markdown": "An angle formed by two chords intersecting within the circumference is measured by half the sum of the intercepted arcs.", "why": "Packages the two-chord rule as one memorable statement that a learner can check against a diagram.", "use": [ "lesson", "website" ], "concepts": [ "concept/chord", "theorem/chord-angle-measured-by-half-the-sum-of-intercepted-arcs" ] }, { "id": "wentworth-plane-geometry-1899/x-816fd5bcb1", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 112", "location": "THE CIRCLE", "latex": "An angle inscribed in a semicircle is a right angle.", "markdown": "An angle inscribed in a semicircle is a right angle.", "why": "A short corollary a learner can recall and use at once, and a quick test of whether the inscribed-angle rule is understood.", "use": [ "lesson", "website" ], "concepts": [ "concept/inscribed-angle", "theorem/angle-inscribed-in-a-semicircle-is-a-right-angle" ] }, { "id": "wentworth-plane-geometry-1899/x-1ea73a8cd8", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 84", "location": "THE CIRCLE", "latex": "A \\textbf{circle}\\label{circle} is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within called the \\textbf{centre}\\label{centrecirc}.", "markdown": "A **circle** is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within called the **centre**.", "why": "Gives the learner the defining property of a circle in one sentence: every point on the boundary is the same distance from the centre.", "use": [ "lesson", "website" ], "concepts": [ "concept/centre-of-a-circle", "concept/circle" ] }, { "id": "wentworth-plane-geometry-1899/x-63260e02da", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 101", "location": "THE CIRCLE", "latex": "Two quantities of the same kind that cannot \\emph{both} be expressed in \\emph{integers} in terms of a common unit, are said to be \\textbf{incommensurable}, and the \\emph{exact value} of their ratio cannot be found.", "markdown": "Two quantities of the same kind that cannot *both* be expressed in *integers* in terms of a common unit, are said to be **incommensurable**, and the *exact value* of their ratio cannot be found.", "why": "Shows that incommensurability means no common unit gives whole-number measures, so the ratio has no exact value.", "use": [ "lesson" ], "concepts": [ "concept/incommensurable-magnitudes" ] }, { "id": "wentworth-plane-geometry-1899/x-06e6bf1984", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 88", "location": "THE CIRCLE", "latex": "If, however, a theorem is in fact a group of three theorems, and if \\emph{one of the hypotheses} of the group \\emph{must} be true, and \\emph{no two of the conclusions can be true at the same time}, then the converse of the theorem is \\emph{necessarily} true.", "markdown": "If, however, a theorem is in fact a group of three theorems, and if *one of the hypotheses* of the group *must* be true, and *no two of the conclusions can be true at the same time*, then the converse of the theorem is *necessarily* true.", "why": "Corrects the common belief that every converse is suspect by showing when a converse is automatically valid.", "use": [ "lesson" ], "concepts": [ "concept/converse-of-a-theorem" ] }, { "id": "wentworth-plane-geometry-1899/x-9d64287eef", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 101", "location": "THE CIRCLE", "latex": "To \\textbf{measure} a quantity of any kind is to find \\emph{the number of times} it contains a known quantity of the \\emph{same kind}, called the \\textbf{unit of measure}.", "markdown": "To **measure** a quantity of any kind is to find *the number of times* it contains a known quantity of the *same kind*, called the **unit of measure**.", "why": "Gives the basic meaning of measurement as counting how many units a quantity contains.", "use": [ "lesson" ], "concepts": [ "unit/unit" ] }, { "id": "wentworth-plane-geometry-1899/x-5a487c1b47", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 95", "location": "THE CIRCLE", "latex": "A straight line perpendicular to a radius at its extremity is a tangent to the circle.", "markdown": "A straight line perpendicular to a radius at its extremity is a tangent to the circle.", "why": "States the test a learner uses to show that a line touches a circle at one point.", "use": [ "lesson" ], "concepts": [ "concept/tangent", "theorem/tangent-perpendicular-to-radius" ] }, { "id": "wentworth-plane-geometry-1899/x-840658d23e", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 84", "location": "THE CIRCLE", "latex": "By the definition of a circle, \\emph{all its radii are equal}. All its diameters are equal, since a diameter is equal to two radii.", "markdown": "By the definition of a circle, *all its radii are equal*. All its diameters are equal, since a diameter is equal to two radii.", "why": "Shows a learner why all radii of one circle are equal, directly from the definition of a circle.", "use": [ "lesson" ], "concepts": [ "concept/circle", "concept/diameter", "quantity/radius" ] }, { "id": "wentworth-plane-geometry-1899/x-d48594ea2f", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 84", "location": "THE CIRCLE", "latex": "A \\indexbf{tangent} is a straight line of unlimited length which has one point, and only one, in common with the circumference; as, $BC$ (Fig.~1).", "markdown": "A **tangent** is a straight line of unlimited length which has one point, and only one, in common with the circumference; as, $BC$ (Fig. 1).", "why": "Gives the one-point-of-contact definition of a tangent that learners need before the tangent theorems.", "use": [ "lesson" ], "concepts": [ "concept/point-of-tangency", "concept/tangent" ] }, { "id": "wentworth-plane-geometry-1899/x-2bdd72ca57", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 101", "location": "THE CIRCLE", "latex": "No quantity is great or small except by comparison with another quantity of the \\emph{same kind}.", "markdown": "No quantity is great or small except by comparison with another quantity of the *same kind*.", "why": "Explains that a ratio only makes sense between quantities of the same kind.", "use": [ "lesson" ], "concepts": [ "concept/quantities-of-the-same-kind", "concept/ratio" ] }, { "id": "wentworth-plane-geometry-1899/x-a9f1c88e27", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 101", "location": "THE CIRCLE", "latex": "But by taking the unit sufficiently small, an \\emph{approximate value} can be found that shall differ from the true value of the ratio by less than any assigned value, however small.", "markdown": "But by taking the unit sufficiently small, an *approximate value* can be found that shall differ from the true value of the ratio by less than any assigned value, however small.", "why": "Explains how an incommensurable ratio is handled in practice by successive approximation.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/incommensurable-magnitudes", "concept/incommensurable-ratio" ] }, { "id": "wentworth-plane-geometry-1899/x-224387316b", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 102", "location": "THE CIRCLE", "latex": "If a variable, by having different successive values, can be made to differ from a given constant by less than any assigned value, however small, but cannot be made absolutely equal to the constant, that constant is called the \\indexbf{limit} of the variable, and the variable is said to \\textbf{approach the constant as its limit}.", "markdown": "If a variable, by having different successive values, can be made to differ from a given constant by less than any assigned value, however small, but cannot be made absolutely equal to the constant, that constant is called the **limit** of the variable, and the variable is said to **approach the constant as its limit**.", "why": "Gives the two-part test for a limit that learners apply to every later limit argument.", "use": [ "lesson" ], "concepts": [ "concept/constant", "concept/limit", "concept/variable" ] }, { "id": "wentworth-plane-geometry-1899/x-7fde54b7df", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 103", "location": "THE CIRCLE", "latex": "Then it is evident that the moving point \\emph{may approach as near to $B$ as we choose, but will never arrive at $B$}.", "markdown": "Then it is evident that the moving point *may approach as near to $B$ as we choose, but will never arrive at $B$*.", "why": "A vivid moving-point picture that makes the idea of a limit that is never reached concrete.", "use": [ "website" ], "concepts": [ "concept/limit", "concept/variable" ] }, { "id": "wentworth-plane-geometry-1899/x-b8cf97efac", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 110", "location": "THE CIRCLE", "latex": "We cannot make $DB'$ equal to zero, since, by hypothesis, $AB$ and $A'B'$ are incommensurable.", "markdown": "We cannot make $DB'$ equal to zero, since, by hypothesis, $AB$ and $A'B'$ are incommensurable.", "why": "It shows the learner why incommensurable magnitudes force a limiting argument rather than a direct count of parts.", "use": [ "lesson" ], "concepts": [ "concept/incommensurable-magnitudes", "concept/limit", "theorem/central-angles-have-the-same-ratio-as-their-intercepted-arcs" ] }, { "id": "wentworth-plane-geometry-1899/x-082dad3dcc", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 115", "location": "THE CIRCLE", "latex": "Thus, if $OA$ is considered positive, then $OC$ may be considered negative, and if $OR$ is considered positive, then $OD$ may be considered negative.", "markdown": "Thus, if $OA$ is considered positive, then $OC$ may be considered negative, and if $OR$ is considered positive, then $OD$ may be considered negative.", "why": "A concrete pair of opposite directions makes the sign convention for signed quantities easy to hold in mind.", "use": [ "lesson" ], "concepts": [ "concept/signed-quantity" ] }, { "id": "wentworth-plane-geometry-1899/x-a4d706b429", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 116", "location": "THE CIRCLE", "latex": "By marking the distinction between quantities measured in opposite directions, a theorem may often be so stated as to include two or more particular theorems.", "markdown": "By marking the distinction between quantities measured in opposite directions, a theorem may often be so stated as to include two or more particular theorems.", "why": "It tells the learner that signs let one theorem cover several cases, which is the point of the Principle of Continuity.", "use": [ "lesson", "history" ], "concepts": [ "concept/signed-quantity", "method/principle-of-continuity" ] }, { "id": "wentworth-plane-geometry-1899/x-d996b3f53e", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 114", "location": "THE CIRCLE", "latex": "An angle included by a tangent and a chord drawn from the point of contact is measured by half the intercepted arc.", "markdown": "An angle included by a tangent and a chord drawn from the point of contact is measured by half the intercepted arc.", "why": "It states in one line the result a learner needs to connect tangents with arcs on a circle.", "use": [ "lesson", "website" ], "concepts": [ "concept/point-of-tangency", "concept/tangent", "theorem/angle-formed-by-a-tangent-and-a-chord-is-measured-by-half-the-intercepted-arc", "theorem/tangent-chord-angle-measured-by-half-its-intercepted-arc" ] }, { "id": "wentworth-plane-geometry-1899/x-e63b24a578", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 116", "location": "THE CIRCLE", "latex": "Here the word \\emph{sum} means the algebraic sum and includes both the arithmetical sum and the arithmetical difference of two quantities.", "markdown": "Here the word *sum* means the algebraic sum and includes both the arithmetical sum and the arithmetical difference of two quantities.", "why": "It warns that 'sum' in a circle theorem can mean a difference, a common stumbling point for beginners.", "use": [ "lesson" ], "concepts": [ "concept/algebraic-sum", "method/principle-of-continuity" ] }, { "id": "wentworth-plane-geometry-1899/x-a71066095f", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 139", "location": "THE CIRCLE", "latex": "Prove that the locus of the vertex of a right triangle, having a given hypotenuse as base, is the circumference described upon the given hypotenuse as diameter (§~290).", "markdown": "Prove that the locus of the vertex of a right triangle, having a given hypotenuse as base, is the circumference described upon the given hypotenuse as diameter (§ 290).", "why": "It shows a locus problem whose answer is a circle, which gives learners a concrete picture of what a locus is.", "use": [ "lesson" ], "concepts": [ "concept/circle", "concept/hypotenuse", "concept/locus", "concept/right-triangle" ] }, { "id": "wentworth-plane-geometry-1899/x-269be65811", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 138", "location": "THE CIRCLE", "latex": "The required point is the intersection of the given line with the perpendicular bisector of the line joining the two given points (§~160).", "markdown": "The required point is the intersection of the given line with the perpendicular bisector of the line joining the two given points (§ 160).", "why": "It states the method for finding a point equidistant from two given points, a standard construction step.", "use": [ "lesson" ], "concepts": [ "concept/construction", "concept/perpendicular-bisector", "quantity/distance" ] }, { "id": "wentworth-plane-geometry-1899/x-3e5d1fc1f4", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 143", "location": "THE CIRCLE", "latex": "Make use of the point which forms with $P$ a pair of points symmetrical with respect to $AB$.", "markdown": "Make use of the point which forms with $P$ a pair of points symmetrical with respect to $AB$.", "why": "It hints that reflecting a point across a line can turn a path problem into a straight-line problem.", "use": [ "lesson" ], "concepts": [ "concept/symmetry", "concept/tangent" ] }, { "id": "wentworth-plane-geometry-1899/x-fe9b4629af", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 143", "location": "THE CIRCLE", "latex": "Let $r$ and $r'$ denote the radii of the circles, $O$ and $O'$ their centres.", "markdown": "Let $r$ and $r'$ denote the radii of the circles, $O$ and $O'$ their centres.", "why": "It sets up two circles with named radii and centres, the notation learners need for the common tangent construction.", "use": [ "lesson" ], "concepts": [ "concept/centre-of-a-circle", "concept/tangent", "quantity/radius" ] }, { "id": "wentworth-plane-geometry-1899/x-dbef2fe938", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 140", "location": "THE CIRCLE", "latex": "Let $ABC$ be the $\\triangle$ required, $EF$ the given perimeter. The altitude $CD$ passes through the middle of $EF$, and the $\\triangle_s AEC$, $BFC$ are isosceles.", "markdown": "Let $ABC$ be the $\\triangle$ required, $EF$ the given perimeter. The altitude $CD$ passes through the middle of $EF$, and the $\\triangle_s AEC$, $BFC$ are isosceles.", "why": "It shows how a given perimeter and altitude fix the triangle through isosceles sub-triangles.", "use": [ "lesson" ], "concepts": [ "concept/altitude-of-a-triangle", "concept/isosceles-triangle", "concept/perimeter" ] }, { "id": "wentworth-plane-geometry-1899/x-726fa69ba5", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 143", "location": "THE CIRCLE", "latex": "To bisect the angle formed by two lines, without producing the lines to their point of intersection.", "markdown": "To bisect the angle formed by two lines, without producing the lines to their point of intersection.", "why": "It poses a classic construction problem that tests whether learners can adapt a method when the usual setup is unavailable.", "use": [ "lesson", "history" ], "concepts": [ "concept/angle-bisector", "concept/construction" ] }, { "id": "wentworth-plane-geometry-1899/x-ba4bdf36d0", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 143", "location": "THE CIRCLE", "latex": "To draw the internal tangents use an auxiliary $\\odot$ of radius $r + r'$.", "markdown": "To draw the internal tangents use an auxiliary $\\odot$ of radius $r + r'$.", "why": "It gives a useful hint that an auxiliary circle can reduce a tangent problem to a simpler one.", "use": [ "lesson" ], "concepts": [ "concept/circle", "concept/tangent", "quantity/radius" ] }, { "id": "wentworth-plane-geometry-1899/x-6e3c52d657", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 144", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "A \\indexbf{proportion} is an expression of equality between two equal ratios; and is written in one of the following forms:", "markdown": "A **proportion** is an expression of equality between two equal ratios; and is written in one of the following forms:", "why": "It states plainly what a proportion is before any theorem about one is used.", "use": [ "lesson" ], "concepts": [ "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/x-544009887d", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 145", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "In every proportion the product of the extremes is equal to the product of the means.", "markdown": "In every proportion the product of the extremes is equal to the product of the means.", "why": "This is the rule a learner can check on any proportion by cross-multiplying.", "use": [ "lesson" ], "concepts": [ "concept/extremes-of-a-proportion", "concept/means-of-a-proportion", "theorem/product-of-extremes-equals-product-of-means" ] }, { "id": "wentworth-plane-geometry-1899/x-cb4b6795d4", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 144", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "If three quantities are in continued proportion, the second is called the \\indexbf{mean proportional} between the other two, and the third is called the \\textbf{third proportional} to the other two.", "markdown": "If three quantities are in continued proportion, the second is called the **mean proportional** between the other two, and the third is called the **third proportional** to the other two.", "why": "It names the mean and third proportionals in terms of a single chain of equal ratios, which makes the naming easy to follow.", "use": [ "lesson", "history" ], "concepts": [ "concept/continued-proportion", "concept/geometrical-mean", "concept/third-proportional" ] }, { "id": "wentworth-plane-geometry-1899/x-1628dabbfa", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 144", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "The fourth proportional to three given quantities is the fourth term of the proportion which has for its first three terms the three given quantities \\emph{taken in order.}", "markdown": "The fourth proportional to three given quantities is the fourth term of the proportion which has for its first three terms the three given quantities *taken in order.*", "why": "The phrase 'taken in order' tells a learner that the position of each given quantity matters.", "use": [ "lesson" ], "concepts": [ "concept/fourth-proportional" ] }, { "id": "wentworth-plane-geometry-1899/x-489e486a75", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 145", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "If the product of two quantities is equal to the product of two others, either two may be made the extremes of the proportion in which the other two are made the means.", "markdown": "If the product of two quantities is equal to the product of two others, either two may be made the extremes of the proportion in which the other two are made the means.", "why": "It shows how a learner can build a proportion from an equation of products, which is the reverse of the cross-multiplication rule.", "use": [ "lesson" ], "concepts": [ "theorem/equal-products-give-a-proportion", "theorem/product-of-extremes-equals-product-of-means" ] }, { "id": "wentworth-plane-geometry-1899/x-04c4105e8a", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 146", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "If four quantities are in proportion, they are in proportion by \\textnormal{\\indexbf{inversion}}; that is, the second term is to the first as the fourth is to the third.", "markdown": "If four quantities are in proportion, they are in proportion by **inversion**; that is, the second term is to the first as the fourth is to the third.", "why": "It gives a reversible move on a proportion, so a learner can see that flipping both ratios keeps the equality.", "use": [ "lesson" ], "concepts": [ "theorem/inversion-of-proportions" ] }, { "id": "wentworth-plane-geometry-1899/x-5a42042395", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 144", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "Thus, in the proportion $a:b = b:c$; $b$ is the mean proportional between $a$ and $c$; and $c$ is the third proportional to $a$ and $b$.", "markdown": "Thus, in the proportion $a:b = b:c$; $b$ is the mean proportional between $a$ and $c$; and $c$ is the third proportional to $a$ and $b$.", "why": "Shows a learner, with a concrete example, how the mean proportional and third proportional sit in a continued proportion.", "use": [ "lesson" ], "concepts": [ "concept/continued-proportion", "concept/geometrical-mean", "concept/third-proportional" ] }, { "id": "wentworth-plane-geometry-1899/x-4d554c39ed", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 145", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "The mean proportional between two quantities is equal to the square root of their product.", "markdown": "The mean proportional between two quantities is equal to the square root of their product.", "why": "Gives the one-line rule a learner needs to compute a mean proportional, and links it to square roots.", "use": [ "lesson" ], "concepts": [ "concept/geometrical-mean", "method/extracting-the-square-root" ] }, { "id": "wentworth-plane-geometry-1899/x-f7c6bd1691", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 150", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "In the treatment of proportion, it is assumed that the \\emph{quantities} involved are expressed by their \\emph{numerical measures}.", "markdown": "In the treatment of proportion, it is assumed that the *quantities* involved are expressed by their *numerical measures*.", "why": "Warns the learner that proportion works on numbers measured in a common unit, not on the quantities themselves.", "use": [ "lesson" ], "concepts": [ "concept/numerical-measure", "concept/quantities-of-the-same-kind", "concept/ratio" ] }, { "id": "wentworth-plane-geometry-1899/x-54b8ecb75b", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 152", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "By increasing the \\emph{number} of equal parts into which $AE$ is divided, we can make the \\emph{length} of each part less than any assigned value, however small, but not zero.", "markdown": "By increasing the *number* of equal parts into which $AE$ is divided, we can make the *length* of each part less than any assigned value, however small, but not zero.", "why": "Gives a clear, vivid explanation of the limit argument that handles incommensurable segments.", "use": [ "lesson", "history" ], "concepts": [ "concept/incommensurable-magnitudes", "concept/limit" ] }, { "id": "wentworth-plane-geometry-1899/x-74a576bd46", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 157", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\indexbf{Similar polygons} are polygons that have their homologous angles equal, and their homologous sides proportional.", "markdown": "**Similar polygons** are polygons that have their homologous angles equal, and their homologous sides proportional.", "why": "States the two conditions of similarity precisely, the core definition a learner must hold.", "use": [ "lesson", "website" ], "concepts": [ "concept/homologous-sides", "concept/mutually-equiangular-polygons", "concept/similar-polygons" ] }, { "id": "wentworth-plane-geometry-1899/x-e95b2632ac", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 157", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "The primary idea of similarity is \\textbf{likeness of form}.", "markdown": "The primary idea of similarity is **likeness of form**.", "why": "A memorable plain-language picture of similarity that a website reader can grasp at once.", "use": [ "website" ], "concepts": [ "concept/similar-polygons", "concept/similarity" ] }, { "id": "wentworth-plane-geometry-1899/x-8670d14d59", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 171", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.", "markdown": "The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.", "why": "It states the pythagorean theorem in one plain sentence a learner can carry into any right-triangle problem.", "use": [ "lesson" ], "concepts": [ "theorem/pythagorean-theorem" ] }, { "id": "wentworth-plane-geometry-1899/x-0030f8cea3", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 184", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "A straight line is divided \\textbf{in extreme and mean ratio}\\label{extrememean}, when one of the segments is the mean proportional between the whole line and the other segment.", "markdown": "A straight line is divided **in extreme and mean ratio**, when one of the segments is the mean proportional between the whole line and the other segment.", "why": "It defines the extreme and mean ratio by naming the mean-proportional relation that makes the division work.", "use": [ "lesson", "website" ], "concepts": [ "concept/extreme-and-mean-ratio", "concept/geometrical-mean" ] }, { "id": "wentworth-plane-geometry-1899/x-6618b16436", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 176", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "If from a fixed point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn.", "markdown": "If from a fixed point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn.", "why": "It shows a surprising constant product that a learner can test by drawing secants from one point.", "use": [ "lesson", "website" ], "concepts": [ "concept/secant" ] }, { "id": "wentworth-plane-geometry-1899/x-782272bdfb", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 169", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "The perpendicular is the mean proportional between the segments of the hypotenuse.", "markdown": "The perpendicular is the mean proportional between the segments of the hypotenuse.", "why": "It gives the key right-triangle fact that links a perpendicular to the two segments it cuts from the hypotenuse.", "use": [ "lesson" ], "concepts": [ "concept/geometrical-mean", "concept/hypotenuse" ] }, { "id": "wentworth-plane-geometry-1899/x-f26d3ad3a2", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 173", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "The last three theorems enable us to compute the lengths of the altitudes of a triangle if the lengths of the three sides are known.", "markdown": "The last three theorems enable us to compute the lengths of the altitudes of a triangle if the lengths of the three sides are known.", "why": "It shows learners why the proportion results matter, since they turn side lengths into altitude lengths.", "use": [ "lesson" ], "concepts": [ "concept/altitude-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/x-c211b0108d", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 171", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "The \\textbf{projection} of any line upon a second line is the segment of the second line included between the perpendiculars drawn to it from the extremities of the first line.", "markdown": "The **projection** of any line upon a second line is the segment of the second line included between the perpendiculars drawn to it from the extremities of the first line.", "why": "It defines projection precisely so that learners can read the later theorems that depend on it.", "use": [ "lesson" ], "concepts": [ "concept/projection" ] }, { "id": "wentworth-plane-geometry-1899/x-cf843dae46", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 175", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "that is, the ratio of two corresponding segments is equal to the \\emph{reciprocal} of the ratio of the other two segments.", "markdown": "that is, the ratio of two corresponding segments is equal to the *reciprocal* of the ratio of the other two segments.", "why": "It restates the intersecting-chords proportion in words, so learners can see the reciprocal relation between two pairs of segments.", "use": [ "lesson" ], "concepts": [ "concept/common-ratio", "concept/reciprocal" ] }, { "id": "wentworth-plane-geometry-1899/x-09627a50d3", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 193", "location": "AREAS OF POLYGONS", "latex": "The \\textbf{area of a surface}\\label{area} is the \\emph{number of units of surface} it contains.", "markdown": "The **area of a surface** is the *number of units of surface* it contains.", "why": "It defines area as a count of unit squares, which is the idea the rest of the book depends on.", "use": [ "lesson", "website" ], "concepts": [ "concept/area" ] }, { "id": "wentworth-plane-geometry-1899/x-337ff1d460", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 193", "location": "AREAS OF POLYGONS", "latex": "Two rectangles having equal altitudes are to each other as their bases.", "markdown": "Two rectangles having equal altitudes are to each other as their bases.", "why": "It states the first result of the chapter in one sentence the learner can carry into the proof.", "use": [ "lesson" ], "concepts": [ "theorem/area-of-rectangles-with-equal-altitudes" ] }, { "id": "wentworth-plane-geometry-1899/x-f933653a17", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 193", "location": "AREAS OF POLYGONS", "latex": "The \\textbf{unit of surface} is a square whose side is a \\emph{unit of length}.", "markdown": "The **unit of surface** is a square whose side is a *unit of length*.", "why": "It fixes the basic unit against which every area in the book is measured.", "use": [ "lesson", "website" ], "concepts": [ "quantity/area", "unit/unit-of-surface" ] }, { "id": "wentworth-plane-geometry-1899/x-19205e1914", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 193", "location": "AREAS OF POLYGONS", "latex": "Plane figures that \\emph{have equal areas but cannot be made to coincide} are called \\textbf{equivalent}\\label{equivalent2}.", "markdown": "Plane figures that *have equal areas but cannot be made to coincide* are called **equivalent**.", "why": "It separates equal area from congruence, a distinction learners often blur.", "use": [ "lesson" ], "concepts": [ "concept/equivalent-plane-figures" ] }, { "id": "wentworth-plane-geometry-1899/x-bb179e78d5", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 193", "location": "AREAS OF POLYGONS", "latex": "In propositions relating to \\emph{areas}, the words ``rectangle,'' ``triangle,'' etc., are often used for ``area of rectangle,'' ``area of triangle,'' etc.", "markdown": "In propositions relating to *areas*, the words “rectangle,” “triangle,” etc., are often used for “area of rectangle,” “area of triangle,” etc.", "why": "It warns the reader that a bare figure name in a proposition means its area, which is easy to misread.", "use": [ "lesson", "history" ], "concepts": [ "concept/area", "concept/rectangle", "concept/triangle", "quantity/area" ] }, { "id": "wentworth-plane-geometry-1899/x-95a089455f", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 196", "location": "AREAS OF POLYGONS", "latex": "The area of a rectangle is equal to the product of its base by its altitude.", "markdown": "The area of a rectangle is equal to the product of its base by its altitude.", "why": "It gives the central formula on which the rest of the chapter is built.", "use": [ "lesson", "website" ], "concepts": [ "concept/rectangle", "theorem/area-of-a-rectangle" ] }, { "id": "wentworth-plane-geometry-1899/x-aa9ed859ce", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 196", "location": "AREAS OF POLYGONS", "latex": "When the base and altitude each contain the linear unit an integral number of times, this proposition is rendered evident by dividing the figure into squares, each equal to the unit of surface.", "markdown": "When the base and altitude each contain the linear unit an integral number of times, this proposition is rendered evident by dividing the figure into squares, each equal to the unit of surface.", "why": "It lets a learner see the rectangle formula as counting unit squares before any proof is read.", "use": [ "lesson" ], "concepts": [ "theorem/area-of-a-rectangle", "unit/unit-of-surface" ] }, { "id": "wentworth-plane-geometry-1899/x-d4e051bbd7", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 199", "location": "AREAS OF POLYGONS", "latex": "The area of an irregular polygon may be found by dividing the polygon into triangles, and by finding the area of each of these triangles separately.", "markdown": "The area of an irregular polygon may be found by dividing the polygon into triangles, and by finding the area of each of these triangles separately.", "why": "It shows how a messy shape reduces to triangles whose areas are already known.", "use": [ "lesson", "website" ], "concepts": [ "concept/polygon", "method/area-of-an-irregular-polygon", "theorem/area-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/x-dd4082ade1", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 203", "location": "AREAS OF POLYGONS", "latex": "The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the two legs.", "markdown": "The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the two legs.", "why": "It states the Pythagorean relation as an equality of areas, which links the chapter to the rest of the geometry.", "use": [ "lesson", "history" ], "concepts": [ "concept/hypotenuse", "concept/leg-of-a-right-triangle", "concept/right-triangle", "theorem/pythagorean-theorem" ] }, { "id": "wentworth-plane-geometry-1899/x-9b3ddfe50a", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 220", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "A \\indexbf{regular polygon} is a polygon which is both equilateral and equiangular. The equilateral triangle and the square are examples.", "markdown": "A **regular polygon** is a polygon which is both equilateral and equiangular. The equilateral triangle and the square are examples.", "why": "Gives the learner the defining pair of conditions for a regular polygon, with familiar examples to anchor it.", "use": [ "lesson", "website" ], "concepts": [ "concept/equilateral-triangle", "concept/regular-polygon", "concept/square" ] }, { "id": "wentworth-plane-geometry-1899/x-7525f46e3b", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 231", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "The ratio of the circumference of a circle to its diameter is constant.", "markdown": "The ratio of the circumference of a circle to its diameter is constant.", "why": "States the fact that makes pi possible, which a learner can check on any circle they draw.", "use": [ "lesson" ], "concepts": [ "concept/pi", "quantity/circumference" ] }, { "id": "wentworth-plane-geometry-1899/x-7c7dd49317", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 231", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "The constant ratio of the circumference of a circle to its diameter is represented by the Greek letter $\\pi$\\label{pi}.", "markdown": "The constant ratio of the circumference of a circle to its diameter is represented by the Greek letter $\\pi$.", "why": "Names pi as the constant ratio, so the symbol means something concrete before any numbers are used.", "use": [ "lesson", "history" ], "concepts": [ "concept/pi" ] }, { "id": "wentworth-plane-geometry-1899/x-1224ccd7bd", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 227", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "If the number of sides of a regular inscribed polygon is indefinitely increased, the apothem of the polygon approaches the radius of the circle as its limit.", "markdown": "If the number of sides of a regular inscribed polygon is indefinitely increased, the apothem of the polygon approaches the radius of the circle as its limit.", "why": "Shows the idea of a limit in a picture a learner can imagine: more sides, closer to the circle.", "use": [ "lesson", "website" ], "concepts": [ "concept/apothem", "concept/inscribed-circle", "concept/limit", "concept/regular-polygon" ] }, { "id": "wentworth-plane-geometry-1899/x-8f37984bcb", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 232", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "The area of a regular polygon is equal to half the product of its apothem by its perimeter.", "markdown": "The area of a regular polygon is equal to half the product of its apothem by its perimeter.", "why": "Gives a compact formula for the area of any regular polygon, reached by splitting it into triangles.", "use": [ "lesson" ], "concepts": [ "concept/apothem", "concept/perimeter", "theorem/area-of-a-regular-polygon" ] }, { "id": "wentworth-plane-geometry-1899/x-d7bb5d452f", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 238", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "Therefore, to inscribe a regular decagon, divide the radius internally in extreme and mean ratio, and apply the greater segment ten times as a chord.", "markdown": "Therefore, to inscribe a regular decagon, divide the radius internally in extreme and mean ratio, and apply the greater segment ten times as a chord.", "why": "A clear step-by-step construction method that a learner can carry out with compass and straightedge.", "use": [ "lesson" ], "concepts": [ "concept/chord", "concept/decagon", "concept/extreme-and-mean-ratio" ] }, { "id": "wentworth-plane-geometry-1899/x-289cb9038e", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 242", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "$\\pi$ is incommensurable.", "markdown": "$\\pi$ is incommensurable.", "why": "Tells the learner that pi cannot be written exactly as a ratio of whole numbers, which explains why it is only ever approximated.", "use": [ "lesson", "history" ], "concepts": [ "concept/incommensurable-magnitudes", "concept/pi" ] }, { "id": "wentworth-plane-geometry-1899/x-d0aa9ea5a4", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 243", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "Among geometrical magnitudes which satisfy given conditions, the \\emph{greatest} is called the \\indexbf{maximum}; and the \\emph{smallest} is called the \\indexbf{minimum}.", "markdown": "Among geometrical magnitudes which satisfy given conditions, the *greatest* is called the **maximum**; and the *smallest* is called the **minimum**.", "why": "It gives a clear two-part definition of the maximum and minimum, which a learner can state back in their own words.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/minimum" ] }, { "id": "wentworth-plane-geometry-1899/x-6754238415", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 243", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "Of all triangles having two given sides, that in which these sides include a right angle is the maximum.", "markdown": "Of all triangles having two given sides, that in which these sides include a right angle is the maximum.", "why": "It states a memorable result that a learner can test on a few triangles with the same two sides.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/triangle", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/x-961e532717", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 243", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "Thus, the diameter of a circle is the maximum among all chords; and the perpendicular is the minimum among all lines drawn to a given line from a given external point.", "markdown": "Thus, the diameter of a circle is the maximum among all chords; and the perpendicular is the minimum among all lines drawn to a given line from a given external point.", "why": "Two familiar figures show that a maximum or minimum can be found in ordinary geometry, which makes the abstract definition concrete.", "use": [ "website", "lesson" ], "concepts": [ "concept/chord", "concept/diameter", "concept/maximum", "concept/minimum", "concept/perpendicular" ] }, { "id": "wentworth-plane-geometry-1899/x-7e930eced8", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 245", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "Hence, \\emph{every} vertex lies on the circumference; that is, the maximum polygon can be inscribed in a semicircle having the undetermined side for a diameter.", "markdown": "Hence, *every* vertex lies on the circumference; that is, the maximum polygon can be inscribed in a semicircle having the undetermined side for a diameter.", "why": "It shows a proof technique: a local maximum condition at each vertex forces a global geometric conclusion.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/regular-polygon", "concept/semicircle" ] }, { "id": "wentworth-plane-geometry-1899/x-bdd2c0f44e", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 243", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\indexbf{Isoperimetric} polygons are polygons which have equal \\newline perimeters.", "markdown": "**Isoperimetric** polygons are polygons which have equal perimeters.", "why": "It introduces the term isoperimetric in one sentence, so a learner can compare polygons of equal perimeter from the start.", "use": [ "lesson" ], "concepts": [ "concept/isoperimetric-polygon", "concept/perimeter" ] } ], "equations": [ { "id": "wentworth-plane-geometry-1899/eq-54d88bc220", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 18", "location": "RECTILINEAR FIGURES", "latex": "AB = AC+CB", "name": null, "statement": "The whole line AB equals the parts AC and CB added together when C lies on AB; lines are added by prolonging them.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "segment AB (the line AB, limited by points A and B)" }, { "unit": null, "symbol": "AC", "meaning": "segment AC" }, { "unit": null, "symbol": "CB", "meaning": "segment CB" } ], "sympy": "Eq(AB, AC + CB)", "physics": false, "states": [], "concepts": [ "concept/line-segment", "concept/sum" ] }, { "id": "wentworth-plane-geometry-1899/eq-0befb31442", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 18", "location": "RECTILINEAR FIGURES", "latex": "AC = AB-CB", "name": null, "statement": "Segment AC equals AB with CB subtracted, obtained by diminishing AB to C.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "segment AC" }, { "unit": null, "symbol": "AB", "meaning": "segment AB" }, { "unit": null, "symbol": "CB", "meaning": "segment CB" } ], "sympy": "Eq(AC, AB - CB)", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/line-segment" ] }, { "id": "wentworth-plane-geometry-1899/eq-d76d0e6458", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 18", "location": "RECTILINEAR FIGURES", "latex": "AC = 2AB", "name": null, "statement": "A line made of two equal segments AB laid end to end is twice AB (a line multiplied by a number).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "segment AC, the multiple of AB" }, { "unit": null, "symbol": "AB", "meaning": "the given segment, with AB = BC" } ], "sympy": "Eq(AC, 2*AB)", "physics": false, "states": [], "concepts": [ "concept/line-segment", "concept/multiple", "method/multiplication" ] }, { "id": "wentworth-plane-geometry-1899/eq-abf75d4f4f", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 18", "location": "RECTILINEAR FIGURES", "latex": "AD = 3AB", "name": null, "statement": "Three equal segments AB laid end to end make the segment AD, equal to three times AB.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "segment AD, a multiple of AB" }, { "unit": null, "symbol": "AB", "meaning": "the given segment" } ], "sympy": "Eq(AD, 3*AB)", "physics": false, "states": [], "concepts": [ "concept/line-segment", "concept/multiple", "method/multiplication" ] }, { "id": "wentworth-plane-geometry-1899/eq-6eca0f5ed9", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 18", "location": "RECTILINEAR FIGURES", "latex": "AE = 4AB", "name": null, "statement": "Four equal segments AB laid end to end make the segment AE, equal to four times AB.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "AE", "meaning": "segment AE, a multiple of AB" }, { "unit": null, "symbol": "AB", "meaning": "the given segment" } ], "sympy": "Eq(AE, 4*AB)", "physics": false, "states": [], "concepts": [ "concept/line-segment", "concept/multiple", "method/multiplication" ] }, { "id": "wentworth-plane-geometry-1899/eq-b1a8924bfa", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 24", "location": "RECTILINEAR FIGURES", "latex": "\\angle ACB = \\angle DEF", "name": null, "statement": "Any two straight angles are equal, shown by superposing one on the other so that their vertices and sides coincide.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ACB", "meaning": "straight angle with vertex C" }, { "unit": null, "symbol": "DEF", "meaning": "straight angle with vertex E" } ], "sympy": "Eq(angle_ACB, angle_DEF)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/plane-angle", "concept/straight-angle", "theorem/all-straight-angles-are-equal" ] }, { "id": "wentworth-plane-geometry-1899/eq-0953b4e9b1", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 28", "location": "RECTILINEAR FIGURES", "latex": "CE = CK", "name": null, "statement": "If a perpendicular CF is folded over and the two oblique lines cut off equal segments FE and FK from the foot, the two lines CE and CK are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "CE", "meaning": "line from C to E on AB" }, { "unit": null, "symbol": "CK", "meaning": "line from C to K on AB" } ], "sympy": "Eq(CE, CK)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/foot-of-a-line", "concept/line-segment", "concept/perpendicular" ] }, { "id": "wentworth-plane-geometry-1899/eq-e31f9e17b2", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 28", "location": "RECTILINEAR FIGURES", "latex": "\\angle FCE = \\angle FCK", "name": null, "statement": "The angles that the equal oblique lines CE and CK make with the perpendicular CF are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "FCE", "meaning": "angle at C between CF and CE" }, { "unit": null, "symbol": "FCK", "meaning": "angle at C between CF and CK" } ], "sympy": "Eq(angle_FCE, angle_FCK)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/perpendicular", "concept/plane-angle" ] }, { "id": "wentworth-plane-geometry-1899/eq-f688e35c9c", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 32", "location": "RECTILINEAR FIGURES", "latex": "OE > OG", "name": null, "statement": "Of two lines from a point in a perpendicular, the one cutting off the more remote (greater) segment from the foot is the longer.", "kind": "result", "symbols": [ { "unit": null, "symbol": "OE", "meaning": "oblique line from O to the more remote point E on AB" }, { "unit": null, "symbol": "OG", "meaning": "oblique line from O to the nearer point G on AB" } ], "sympy": "Gt(OE, OG)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/line-segment", "concept/perpendicular" ] }, { "id": "wentworth-plane-geometry-1899/eq-c2c8b51f73", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 38", "location": "RECTILINEAR FIGURES", "latex": "\\angle BHK + \\angle HKD = \\text{a st.\\ }\\angle", "name": null, "statement": "Two interior angles on the same side of a transversal cut by parallel lines add to a straight angle, so they are supplementary.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BHK", "meaning": "interior angle at H between HB and HK" }, { "unit": null, "symbol": "HKD", "meaning": "interior angle at K between KH and KD" } ], "sympy": "Eq(angle_BHK + angle_HKD, pi)", "physics": false, "states": [], "concepts": [ "concept/parallel-lines", "concept/plane-angle", "concept/straight-angle", "concept/supplementary-angles", "concept/transversal" ] }, { "id": "wentworth-plane-geometry-1899/eq-c4cadae58d", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 41", "location": "RECTILINEAR FIGURES", "latex": "\\angle A+\\angle B+\\angle BCA = 2", "name": null, "statement": "The three angles of a triangle together equal two right angles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle A of triangle ABC (interior angle at vertex A)" }, { "unit": null, "symbol": "B", "meaning": "angle B of triangle ABC (interior angle at vertex B)" }, { "unit": null, "symbol": "BCA", "meaning": "angle BCA of triangle ABC (interior angle at vertex C)" } ], "sympy": "Eq(A + B + BCA, 2)", "physics": false, "states": [], "concepts": [ "concept/sum", "concept/triangle", "quantity/angle", "quantity/right-angle", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-934ea52728", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 42", "location": "RECTILINEAR FIGURES", "latex": "AB + BC > AC", "name": null, "statement": "In a triangle, the sum of two sides is greater than the third side.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "length of side AB of triangle ABC" }, { "unit": null, "symbol": "BC", "meaning": "length of side BC of triangle ABC" }, { "unit": null, "symbol": "AC", "meaning": "length of side AC of triangle ABC (the longest side in the proof)" } ], "sympy": "AB + BC > AC", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/side", "concept/sum", "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-bd03195dd6", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 42", "location": "RECTILINEAR FIGURES", "latex": "AC - BC < AB", "name": null, "statement": "In a triangle, the difference of two sides is less than the third side.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "length of side AC of triangle ABC" }, { "unit": null, "symbol": "BC", "meaning": "length of side BC of triangle ABC" }, { "unit": null, "symbol": "AB", "meaning": "length of side AB of triangle ABC" } ], "sympy": "AC - BC < AB", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/inequality", "concept/side", "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-2333a4036b", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 42", "location": "RECTILINEAR FIGURES", "latex": "AB > AC", "name": null, "statement": "In triangle ACB, if angle C is greater than angle B, then side AB is greater than side AC (the greater angle lies opposite the greater side).", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "length of side AB of triangle ABC" }, { "unit": null, "symbol": "AC", "meaning": "length of side AC of triangle ABC" } ], "sympy": "AB > AC", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/side", "concept/triangle", "quantity/angle", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-93a402e5e4", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 61", "location": "RECTILINEAR FIGURES", "latex": "AO = OC", "name": null, "statement": "The diagonals of a parallelogram bisect each other: the segments of diagonal AC from A and from C to the intersection point O are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AO", "meaning": "segment of diagonal AC from vertex A to the intersection point O" }, { "unit": null, "symbol": "OC", "meaning": "segment of diagonal AC from O to vertex C" } ], "sympy": "Eq(AO, OC)", "physics": false, "states": [], "concepts": [ "concept/diagonal", "concept/parallelogram", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-2bdaf7feae", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 61", "location": "RECTILINEAR FIGURES", "latex": "BO = OE", "name": null, "statement": "The diagonals of a parallelogram bisect each other: the segments of diagonal BE from B and from E to the intersection point O are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BO", "meaning": "segment of diagonal BE from vertex B to the intersection point O" }, { "unit": null, "symbol": "OE", "meaning": "segment of diagonal BE from O to vertex E" } ], "sympy": "Eq(BO, OE)", "physics": false, "states": [], "concepts": [ "concept/diagonal", "concept/parallelogram", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-1f81b4de1f", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 58", "location": "RECTILINEAR FIGURES", "latex": "BC = AE", "name": null, "statement": "In a parallelogram, one pair of opposite sides is equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BC", "meaning": "length of side BC of parallelogram ABCE" }, { "unit": null, "symbol": "AE", "meaning": "length of the opposite side AE of parallelogram ABCE" } ], "sympy": "Eq(BC, AE)", "physics": false, "states": [], "concepts": [ "concept/parallelogram", "theorem/opposite-sides-of-a-parallelogram-are-equal", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-1f0cb18888", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 58", "location": "RECTILINEAR FIGURES", "latex": "AB = EC", "name": null, "statement": "In a parallelogram, the other pair of opposite sides is equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "length of side AB of parallelogram ABCE" }, { "unit": null, "symbol": "EC", "meaning": "length of the opposite side EC of parallelogram ABCE" } ], "sympy": "Eq(AB, EC)", "physics": false, "states": [], "concepts": [ "concept/parallelogram", "theorem/opposite-sides-of-a-parallelogram-are-equal", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-2497f58cdc", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 63", "location": "RECTILINEAR FIGURES", "latex": "AB = BC = CD", "name": null, "statement": "If parallel lines cut off equal parts on one transversal, they cut off equal parts on every transversal: the segments AB, BC, CD on transversal AD are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "segment of transversal AD between the first and second parallels" }, { "unit": null, "symbol": "BC", "meaning": "segment of transversal AD between the second and third parallels" }, { "unit": null, "symbol": "CD", "meaning": "segment of transversal AD between the third and fourth parallels" } ], "sympy": "And(Eq(AB, BC), Eq(BC, CD))", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/parallel-lines", "concept/transversal", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-989cdce328", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 64", "location": "RECTILINEAR FIGURES", "latex": "BF=FC = \\frac{1}{2}BC", "name": null, "statement": "The line through the midpoints of two sides of a triangle, drawn parallel to the third side, bisects the third side, so BF and FC are each half of BC.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BF", "meaning": "segment of side BC from B to the midpoint F" }, { "unit": null, "symbol": "FC", "meaning": "segment of side BC from the midpoint F to C" }, { "unit": null, "symbol": "BC", "meaning": "length of side BC of triangle ABC" } ], "sympy": "And(Eq(BF, FC), Eq(FC, Rational(1,2)*BC))", "physics": false, "states": [], "concepts": [ "concept/midpoint", "concept/parallel-lines", "concept/side", "concept/triangle", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-089f849d62", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 64", "location": "RECTILINEAR FIGURES", "latex": "DE = BF = \\frac{1}{2}BC", "name": null, "statement": "The segment joining the midpoints of two sides of a triangle equals half the third side.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DE", "meaning": "segment joining the midpoints D of AB and E of AC in triangle ABC" }, { "unit": null, "symbol": "BF", "meaning": "segment of side BC from B to the midpoint F (equal to DE)" }, { "unit": null, "symbol": "BC", "meaning": "length of side BC of triangle ABC" } ], "sympy": "And(Eq(DE, BF), Eq(BF, Rational(1,2)*BC))", "physics": false, "states": [], "concepts": [ "concept/midpoint", "concept/parallel-lines", "concept/side", "concept/triangle", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "wentworth-plane-geometry-1899/eq-7db4069704", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 64", "location": "RECTILINEAR FIGURES", "latex": "\\frac{1}{2} (AB + DC)", "name": null, "statement": "The median of a trapezoid (the line joining the midpoints of its legs) is parallel to the bases and equal to half the sum of the bases; the book states the length as one half of AB plus DC.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "length of one base of the trapezoid" }, { "unit": null, "symbol": "DC", "meaning": "length of the other base of the trapezoid" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/base-of-a-pyramid", "concept/median-of-a-trapezoid", "concept/sum", "concept/trapezoid" ] }, { "id": "wentworth-plane-geometry-1899/eq-40d487f911", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 67", "location": "RECTILINEAR FIGURES", "latex": "(n-2)2", "name": null, "statement": "The sum of the interior angles of a polygon of n sides equals (n-2) times two right angles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of sides of the polygon" }, { "unit": "right angle", "symbol": "S", "meaning": "sum of the interior angles of the polygon (written in the book as the right-angle expression (n-2)2)" } ], "sympy": "Eq(S, (n-2)*2)", "physics": false, "states": [], "concepts": [ "concept/polygon", "quantity/right-angle", "theorem/sum-of-the-interior-angles-of-a-polygon" ] }, { "id": "wentworth-plane-geometry-1899/eq-d6c8e01f32", "chapter": "wentworth-plane-geometry-1899/ch-i", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 67", "location": "RECTILINEAR FIGURES", "latex": "\\displaystyle \\frac{2(n-2)}{n}", "name": null, "statement": "Each angle of an equiangular polygon of n sides equals 2(n-2)/n right angles.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of sides of the equiangular polygon" }, { "unit": "right angle", "symbol": "A", "meaning": "each interior angle of the equiangular polygon" } ], "sympy": "Eq(A, 2*(n-2)/n)", "physics": false, "states": [], "concepts": [ "concept/equiangular-polygon", "concept/regular-polygon", "quantity/right-angle", "theorem/sum-of-the-interior-angles-of-a-polygon" ] }, { "id": "wentworth-plane-geometry-1899/eq-f2b1d13523", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 101", "location": "THE CIRCLE", "latex": "\\dfrac{a}{b}", "name": null, "statement": "The ratio of a to b is written a : b, or as the fraction a/b.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "first of two quantities of the same kind" }, { "unit": null, "symbol": "b", "meaning": "second of two quantities of the same kind" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/quantities-of-the-same-kind" ] }, { "id": "wentworth-plane-geometry-1899/eq-3bad41cdf3", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 104", "location": "THE CIRCLE", "latex": "\\frac{3}{10} + \\frac{3}{100} + \\frac{3}{1000} + \\cdots", "name": null, "statement": "The decimal 0.333... is written as the infinite sum of the fractions 3/10, 3/100, 3/1000, and so on, whose sum approaches 1/3 as a limit.", "kind": "formula", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/decimal-fraction", "concept/infinite-sequence", "concept/limit" ] }, { "id": "wentworth-plane-geometry-1899/eq-6d370a6549", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 102", "location": "THE CIRCLE", "latex": "\\sqrt{2} = 1.41421356\\cdots", "name": null, "statement": "The square root of 2 is an incommensurable value whose decimal expansion is given to eight places, so approximate values can be taken from it.", "kind": "approximation", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/incommensurable-ratio", "concept/root" ] }, { "id": "wentworth-plane-geometry-1899/eq-5597d2b614", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 105", "location": "THE CIRCLE", "latex": "\\dfrac{x}{k} = \\dfrac{1}{k} × x", "name": null, "statement": "Dividing a variable by a finite constant k is the same as multiplying it by 1/k.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "k", "meaning": "finite constant" } ], "sympy": "Eq(x/k, x*(1/k))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/limit", "concept/quotient", "concept/variable" ] }, { "id": "wentworth-plane-geometry-1899/eq-202e463938", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 104", "location": "THE CIRCLE", "latex": "kx=0", "name": null, "statement": "If the limit of the variable x is zero, then the limit of kx, for any finite constant k, is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "finite constant" }, { "unit": null, "symbol": "x", "meaning": "variable whose limit is zero" } ], "sympy": "Eq(k*x, 0)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/limit", "concept/product", "concept/variable" ] }, { "id": "wentworth-plane-geometry-1899/eq-cef083168f", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 105", "location": "THE CIRCLE", "latex": "kx = ka", "name": null, "statement": "The limit of kx equals k times the limit a of x, for any finite constant k.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "finite constant" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "a", "meaning": "limit of the variable x" } ], "sympy": "Eq(k*x, k*a)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/limit", "concept/product", "concept/variable" ] }, { "id": "wentworth-plane-geometry-1899/eq-3113b2ca4c", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 106", "location": "THE CIRCLE", "latex": "xy = ab", "name": null, "statement": "The limit of the product xy of two variables is the product ab of their respective limits, provided neither limit is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" }, { "unit": null, "symbol": "a", "meaning": "limit of x" }, { "unit": null, "symbol": "b", "meaning": "limit of y" } ], "sympy": "Eq(x*y, a*b)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/product", "concept/variable" ] }, { "id": "wentworth-plane-geometry-1899/eq-5a9c3a850f", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 106", "location": "THE CIRCLE", "latex": "x^n = a^n", "name": null, "statement": "The limit of the nth power of a variable x is the nth power a^n of its limit a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "a", "meaning": "limit of the variable x" }, { "unit": null, "symbol": "n", "meaning": "positive integer exponent" } ], "sympy": "Eq(x**n, a**n)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/power", "concept/variable" ] }, { "id": "wentworth-plane-geometry-1899/eq-65ddeae06e", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 105", "location": "THE CIRCLE", "latex": "d+d'+d''+\\cdots < nd", "name": null, "statement": "The sum of n differences is less than n times the largest difference d, which is the step used to show that the sum of the differences can be made smaller than any assigned quantity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "d", "meaning": "largest of the differences between the variables and their limits" }, { "unit": null, "symbol": "d'", "meaning": "difference between the second variable and its limit" }, { "unit": null, "symbol": "d''", "meaning": "difference between the third variable and its limit" }, { "unit": null, "symbol": "n", "meaning": "number of differences" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/limit", "concept/sum" ] }, { "id": "wentworth-plane-geometry-1899/eq-946022f27e", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 107", "location": "THE CIRCLE", "latex": "\\dfrac{a}{b} = r", "name": null, "statement": "If the variables x and y have a constant ratio r and their limits a and b are not zero, then the ratio of the limits a/b equals r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "limit of x" }, { "unit": null, "symbol": "b", "meaning": "limit of y" }, { "unit": null, "symbol": "r", "meaning": "constant ratio of the variables x and y" } ], "sympy": "Eq(a/b, r)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/constant", "concept/limit" ] }, { "id": "wentworth-plane-geometry-1899/eq-05728c061c", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 98", "location": "THE CIRCLE", "latex": "AB = AC", "name": null, "statement": "The two tangents drawn from an external point A to a circle are equal in length.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "tangent segment from A to the point of contact B" }, { "unit": null, "symbol": "AC", "meaning": "tangent segment from A to the point of contact C" } ], "sympy": "Eq(AB, AC)", "physics": false, "states": [], "concepts": [ "concept/point-of-tangency", "concept/tangent" ] }, { "id": "wentworth-plane-geometry-1899/eq-6d98059ba0", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 110", "location": "THE CIRCLE", "latex": "\\dfrac{\\angle A'C'B'}{\\angle ACB} = \\dfrac{\\arc A'B'}{\\arc AB}", "name": null, "statement": "In equal or the same circle, the ratio of two central angles equals the ratio of their intercepted arcs.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A'C'B'", "meaning": "central angle at C' intercepting arc A'B'" }, { "unit": null, "symbol": "ACB", "meaning": "central angle at C intercepting arc AB" }, { "unit": null, "symbol": "arc A'B'", "meaning": "arc intercepted by the angle A'C'B'" }, { "unit": null, "symbol": "arc AB", "meaning": "arc intercepted by the angle ACB" } ], "sympy": "Eq(Ang_A1C1B1/Ang_ACB, Arc_A1B1/Arc_AB)", "physics": false, "states": [], "concepts": [ "concept/central-angle", "concept/commensurable-magnitudes", "concept/common-ratio", "concept/incommensurable-magnitudes", "concept/limit", "quantity/arc-of-a-circle", "theorem/central-angles-have-the-same-ratio-as-their-intercepted-arcs" ] }, { "id": "wentworth-plane-geometry-1899/eq-a58cb94bbe", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 137", "location": "THE CIRCLE", "latex": "\\angle ECF = 90° + \\frac{1}{2}\\angle ACB", "name": null, "statement": "The angle ECF in the analysed triangle equals a right angle plus half the given angle ACB.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "ECF", "meaning": "angle at C in triangle ECF" }, { "unit": "degree", "symbol": "ACB", "meaning": "the given angle at the vertex C of the required triangle" } ], "sympy": "Eq(ECF, 90 + ACB/2)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-triangle", "concept/triangle", "quantity/angle", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/eq-84fc1b0f14", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 137", "location": "THE CIRCLE", "latex": "\\angle E+\\angle F+\\frac{1}{2}\\angle ACB = 90°", "name": null, "statement": "The angles at E and F of triangle ECF, plus half the given angle ACB, sum to a right angle.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "E", "meaning": "angle at E of triangle ECF" }, { "unit": "degree", "symbol": "F", "meaning": "angle at F of triangle ECF" }, { "unit": "degree", "symbol": "ACB", "meaning": "the given angle at the vertex C of the required triangle" } ], "sympy": "Eq(E + F + ACB/2, 90)", "physics": false, "states": [], "concepts": [ "concept/triangle", "quantity/angle", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/eq-3e18ee3c7e", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 137", "location": "THE CIRCLE", "latex": "\\angle E+\\angle F = 90° - \\frac{1}{2}\\angle ACB", "name": null, "statement": "The sum of the angles at E and F equals a right angle minus half the given angle ACB.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "E", "meaning": "angle at E of triangle ECF" }, { "unit": "degree", "symbol": "F", "meaning": "angle at F of triangle ECF" }, { "unit": "degree", "symbol": "ACB", "meaning": "the given angle at the vertex C of the required triangle" } ], "sympy": "Eq(E + F, 90 - ACB/2)", "physics": false, "states": [], "concepts": [ "concept/triangle", "quantity/angle", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/eq-6fc133375f", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 142", "location": "THE CIRCLE", "latex": "EG = FC = \\frac{1}{2}DC", "name": null, "statement": "In the isosceles trapezoid construction, the segments EG and FC are each half of the base DC, as the proof states.", "kind": "result", "symbols": [ { "unit": null, "symbol": "EG", "meaning": "segment EG, cut off on the base by the line CG drawn parallel to FE" }, { "unit": null, "symbol": "FC", "meaning": "segment FC of the trapezoid's base" }, { "unit": null, "symbol": "DC", "meaning": "the base DC of the trapezoid" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bisector", "concept/diameter", "concept/isosceles-trapezoid", "concept/parallel-lines", "concept/trapezoid" ] }, { "id": "wentworth-plane-geometry-1899/eq-b9723b04a2", "chapter": "wentworth-plane-geometry-1899/ch-ii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 141", "location": "THE CIRCLE", "latex": "\\angle BAC = \\angle BCA = 45°", "name": null, "statement": "In the square construction, the angles BAC and BCA each equal 45°, because the triangles ABC and ABE are isosceles and CA is a diagonal of the square.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "BAC", "meaning": "angle at vertex A between sides AB and AC of the square ABCD" }, { "unit": "degree", "symbol": "BCA", "meaning": "angle at vertex C between sides CB and CA of the square ABCD" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/diagonal", "concept/isosceles-triangle", "concept/square", "quantity/angle", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/eq-ca50a6b923", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 144", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "a:b = c:d", "name": null, "statement": "Four quantities form a proportion when the ratio of the first to the second equals the ratio of the third to the fourth.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "first term of the proportion" }, { "unit": null, "symbol": "b", "meaning": "second term of the proportion" }, { "unit": null, "symbol": "c", "meaning": "third term of the proportion" }, { "unit": null, "symbol": "d", "meaning": "fourth term of the proportion" } ], "sympy": "Eq(a/b, c/d)", "physics": false, "states": [], "concepts": [ "concept/antecedent-of-a-proportion", "concept/common-ratio", "concept/consequent-of-a-proportion", "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/eq-4762affdaf", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 144", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "a:b = b:c = c:d = d:e", "name": null, "statement": "Quantities a, b, c, d, e are in continued proportion when each consecutive pair has the same ratio.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "first quantity in the continued proportion" }, { "unit": null, "symbol": "b", "meaning": "second quantity in the continued proportion" }, { "unit": null, "symbol": "c", "meaning": "third quantity in the continued proportion" }, { "unit": null, "symbol": "d", "meaning": "fourth quantity in the continued proportion" }, { "unit": null, "symbol": "e", "meaning": "fifth quantity in the continued proportion" } ], "sympy": "Eq(a/b, b/c)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/continued-proportion", "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/eq-8cb65d4ce0", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 148", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "a+c+e+g : b+d+f+h = a:b", "name": null, "statement": "In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "antecedent of the first ratio" }, { "unit": null, "symbol": "b", "meaning": "consequent of the first ratio" }, { "unit": null, "symbol": "c", "meaning": "antecedent of the second ratio" }, { "unit": null, "symbol": "d", "meaning": "consequent of the second ratio" }, { "unit": null, "symbol": "e", "meaning": "antecedent of the third ratio" }, { "unit": null, "symbol": "f", "meaning": "consequent of the third ratio" }, { "unit": null, "symbol": "g", "meaning": "antecedent of the fourth ratio" }, { "unit": null, "symbol": "h", "meaning": "consequent of the fourth ratio" } ], "sympy": "Eq((a+c+e+g)/(b+d+f+h), a/b)", "physics": false, "states": [], "concepts": [ "concept/antecedent-of-a-proportion", "concept/common-ratio", "concept/consequent-of-a-proportion", "concept/proportion", "concept/sum" ] }, { "id": "wentworth-plane-geometry-1899/eq-f857efa993", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 152", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{EB}{AE} = \\dfrac{FC}{AF}", "name": null, "statement": "In the incommensurable case, by a limiting argument, the ratio EB to AE equals the ratio FC to AF.", "kind": "result", "symbols": [ { "unit": null, "symbol": "EB", "meaning": "length of segment EB" }, { "unit": null, "symbol": "AE", "meaning": "length of segment AE" }, { "unit": null, "symbol": "FC", "meaning": "length of segment FC" }, { "unit": null, "symbol": "AF", "meaning": "length of segment AF" } ], "sympy": "Eq(EB/AE, FC/AF)", "physics": false, "states": [], "concepts": [ "concept/incommensurable-magnitudes", "concept/limit", "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/eq-e74f8c6099", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 159", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{AB}{A'B'}", "name": null, "statement": "placeholder", "kind": "definition", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [] }, { "id": "wentworth-plane-geometry-1899/eq-aa03064f98", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 157", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "AB:A'B' = BC:B'C' = CD:C'D'", "name": null, "statement": "Similar polygons have their homologous angles equal and their homologous sides proportional, so the ratios of corresponding sides are equal.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "side of polygon ABCDE" }, { "unit": null, "symbol": "A'B'", "meaning": "homologous side of polygon A'B'C'D'E'" }, { "unit": null, "symbol": "BC", "meaning": "side of polygon ABCDE" }, { "unit": null, "symbol": "B'C'", "meaning": "homologous side of polygon A'B'C'D'E'" }, { "unit": null, "symbol": "CD", "meaning": "side of polygon ABCDE" }, { "unit": null, "symbol": "C'D'", "meaning": "homologous side of polygon A'B'C'D'E'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/homologous-sides", "concept/proportion", "concept/similar-polygons", "concept/similarity" ] }, { "id": "wentworth-plane-geometry-1899/eq-3f9e0f71e4", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 158", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "AB:A'B' = AC:A'C' = BC:B'C'", "name": null, "statement": "Two mutually equiangular triangles are similar, so their corresponding sides are proportional.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "side of triangle ABC" }, { "unit": null, "symbol": "A'B'", "meaning": "corresponding side of triangle A'B'C'" }, { "unit": null, "symbol": "AC", "meaning": "side of triangle ABC" }, { "unit": null, "symbol": "A'C'", "meaning": "corresponding side of triangle A'B'C'" }, { "unit": null, "symbol": "BC", "meaning": "side of triangle ABC" }, { "unit": null, "symbol": "B'C'", "meaning": "corresponding side of triangle A'B'C'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/homologous-sides", "concept/mutually-equiangular-polygons", "concept/proportion", "concept/similar-triangles" ] }, { "id": "wentworth-plane-geometry-1899/eq-17fe6592bf", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 162", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac {CO}{C'O'}=\\dfrac {AC}{A'C'}=\\dfrac {AB}{A'B'}=\\dfrac {BC}{B'C'}", "name": null, "statement": "The homologous altitudes of two similar triangles have the same ratio as any two homologous sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "CO", "meaning": "altitude of triangle ABC from C" }, { "unit": null, "symbol": "C'O'", "meaning": "homologous altitude of triangle A'B'C' from C'" }, { "unit": null, "symbol": "AC", "meaning": "side of triangle ABC" }, { "unit": null, "symbol": "A'C'", "meaning": "homologous side of triangle A'B'C'" }, { "unit": null, "symbol": "AB", "meaning": "side of triangle ABC" }, { "unit": null, "symbol": "A'B'", "meaning": "homologous side of triangle A'B'C'" }, { "unit": null, "symbol": "BC", "meaning": "side of triangle ABC" }, { "unit": null, "symbol": "B'C'", "meaning": "homologous side of triangle A'B'C'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-triangle", "concept/homologous-sides", "concept/proportion", "concept/similar-triangles" ] }, { "id": "wentworth-plane-geometry-1899/eq-da3be39f6e", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 163", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{AB}{A'B'}= \\dfrac{BC}{B'C'}= \\dfrac{CD}{C'D'}= \\dfrac{DE}{D'E'}", "name": null, "statement": "Two parallels cut by three or more transversals through one point have proportional corresponding segments.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "segment of the first parallel between transversals OA and OB" }, { "unit": null, "symbol": "A'B'", "meaning": "corresponding segment of the second parallel" }, { "unit": null, "symbol": "BC", "meaning": "segment of the first parallel between transversals OB and OC" }, { "unit": null, "symbol": "B'C'", "meaning": "corresponding segment of the second parallel" }, { "unit": null, "symbol": "CD", "meaning": "segment of the first parallel between transversals OC and OD" }, { "unit": null, "symbol": "C'D'", "meaning": "corresponding segment of the second parallel" }, { "unit": null, "symbol": "DE", "meaning": "segment of the first parallel between transversals OD and OE" }, { "unit": null, "symbol": "D'E'", "meaning": "corresponding segment of the second parallel" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/line-segment", "concept/parallel-lines", "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/eq-ad4a6ec4d9", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 170", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\overline{AC}^2=AB × AF", "name": null, "statement": "In a right triangle, the square of a leg equals the hypotenuse times the segment of the hypotenuse adjacent to that leg.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "a leg of the right triangle ABC (length of the side AC)" }, { "unit": null, "symbol": "AB", "meaning": "the hypotenuse of the right triangle ABC" }, { "unit": null, "symbol": "AF", "meaning": "the segment of the hypotenuse adjacent to A, cut off by the perpendicular CF from the right angle" } ], "sympy": "Eq(AC**2, AB*AF)", "physics": false, "states": [], "concepts": [ "concept/hypotenuse", "concept/leg-of-a-right-triangle", "concept/proportion", "concept/right-triangle", "concept/similar-polygons" ] }, { "id": "wentworth-plane-geometry-1899/eq-e0d3c9d91a", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 170", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{\\overline{AC}^2}{\\overline{BC}^2} = \\dfrac{AB × AF}{AB × BF} = \\dfrac{AF}{BF}", "name": null, "statement": "The squares of the two legs of a right triangle are proportional to the segments of the hypotenuse adjacent to them.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "a leg of the right triangle ABC" }, { "unit": null, "symbol": "BC", "meaning": "the other leg of the right triangle ABC" }, { "unit": null, "symbol": "AF", "meaning": "segment of the hypotenuse adjacent to A" }, { "unit": null, "symbol": "BF", "meaning": "segment of the hypotenuse adjacent to B" }, { "unit": null, "symbol": "AB", "meaning": "the hypotenuse" } ], "sympy": "Eq(AC**2/BC**2, AF/BF)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/hypotenuse", "concept/leg-of-a-right-triangle", "concept/proportion", "concept/right-triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-1e6f8875cf", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 170", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{\\overline{AB}^2}{\\overline{AC}^2} = \\dfrac{AB × AB}{AB × AF} = \\dfrac{AB}{AF}", "name": null, "statement": "The square of the hypotenuse over the square of a leg equals the hypotenuse over the segment of the hypotenuse adjacent to that leg.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "the hypotenuse of the right triangle ABC" }, { "unit": null, "symbol": "AC", "meaning": "a leg of the right triangle ABC" }, { "unit": null, "symbol": "AF", "meaning": "segment of the hypotenuse adjacent to A" } ], "sympy": "Eq(AB**2/AC**2, AB/AF)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/hypotenuse", "concept/leg-of-a-right-triangle", "concept/proportion", "concept/right-triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-498ab4bcdc", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 171", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "AB(AF + BF) = \\overline{AB}^2", "name": "pythagorean theorem", "statement": "The squares of the two legs of a right triangle sum to the square of the hypotenuse; this is the closing algebraic step of the proof (the sum of squares is written across two braces in the source).", "kind": "law", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "the hypotenuse of the right triangle ABC" }, { "unit": null, "symbol": "AF", "meaning": "segment of the hypotenuse adjacent to A" }, { "unit": null, "symbol": "BF", "meaning": "segment of the hypotenuse adjacent to B" } ], "sympy": "Eq(AB*(AF + BF), AB**2)", "physics": false, "states": [ "theorem/pythagorean-theorem" ], "concepts": [ "concept/hypotenuse", "concept/leg-of-a-right-triangle", "concept/right-triangle", "concept/square", "concept/sum" ] }, { "id": "wentworth-plane-geometry-1899/eq-1c7555f123", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 171", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\overline{AC}^2 = \\overline{AB}^2 + \\overline{BC}^2 = 2 \\overline{AB}^2", "name": null, "statement": "For a square ABCD, the square of the diagonal AC equals twice the square of a side AB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "diagonal of the square" }, { "unit": null, "symbol": "AB", "meaning": "a side of the square" }, { "unit": null, "symbol": "BC", "meaning": "a side of the square" } ], "sympy": "Eq(AC**2, 2*AB**2)", "physics": false, "states": [], "concepts": [ "concept/diagonal", "concept/incommensurable-magnitudes", "concept/square", "theorem/pythagorean-theorem" ] }, { "id": "wentworth-plane-geometry-1899/eq-4192b4dc1f", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 171", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "AC = AB \\sqrt{2}", "name": null, "statement": "The diagonal of a square is the side times the square root of two, so diagonal and side are incommensurable.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "diagonal of the square" }, { "unit": null, "symbol": "AB", "meaning": "a side of the square" } ], "sympy": "Eq(AC, AB*sqrt(2))", "physics": false, "states": [], "concepts": [ "concept/diagonal", "concept/incommensurable-magnitudes", "concept/root", "concept/square" ] }, { "id": "wentworth-plane-geometry-1899/eq-cf8a09db25", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 172", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\overline{AB}^2 = \\overline{BC}^2 + \\overline{AC}^2 - 2 BC × DC", "name": null, "statement": "In any triangle, the square of the side opposite an acute angle C equals the sum of the squares of the other two sides minus twice the product of one of them and the projection of the other upon it.", "kind": "law", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "side of triangle ABC opposite the acute angle C" }, { "unit": null, "symbol": "BC", "meaning": "a side of triangle ABC adjacent to C" }, { "unit": null, "symbol": "AC", "meaning": "the other side of triangle ABC adjacent to C" }, { "unit": null, "symbol": "DC", "meaning": "projection of AC upon BC" } ], "sympy": "Eq(AB**2, BC**2 + AC**2 - 2*BC*DC)", "physics": false, "states": [], "concepts": [ "concept/acute-angle", "concept/projection", "concept/triangle", "theorem/pythagorean-theorem" ] }, { "id": "wentworth-plane-geometry-1899/eq-ebf9e51630", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 173", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\overline{AB}^2 = \\overline{BC}^2 + \\overline{AC}^2 + 2 BC × DC", "name": null, "statement": "In an obtuse triangle, the square of the side opposite the obtuse angle C equals the sum of the squares of the other two sides plus twice the product of one of them and the projection of the other upon it.", "kind": "law", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "side of triangle ABC opposite the obtuse angle C" }, { "unit": null, "symbol": "BC", "meaning": "a side of triangle ABC adjacent to C" }, { "unit": null, "symbol": "AC", "meaning": "the other side of triangle ABC adjacent to C" }, { "unit": null, "symbol": "DC", "meaning": "projection of AC upon BC produced" } ], "sympy": "Eq(AB**2, BC**2 + AC**2 + 2*BC*DC)", "physics": false, "states": [], "concepts": [ "concept/obtuse-angle", "concept/projection", "concept/triangle", "theorem/pythagorean-theorem" ] }, { "id": "wentworth-plane-geometry-1899/eq-016b3e0d9e", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 189", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "a^2+b^2 = 2m^2+2\\left(\\dfrac{c}{2}\\right)^2", "name": null, "statement": "In a triangle with sides a, b, c and median m drawn to side c, the sum of the squares of a and b equals twice the square of half of c plus twice the square of the median.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a side of the triangle" }, { "unit": null, "symbol": "b", "meaning": "a side of the triangle" }, { "unit": null, "symbol": "c", "meaning": "the side to which the median is drawn" }, { "unit": null, "symbol": "m", "meaning": "the median to side c" } ], "sympy": "Eq(a**2 + b**2, 2*m**2 + 2*(c/2)**2)", "physics": false, "states": [], "concepts": [ "concept/median-of-a-triangle", "concept/square", "concept/sum", "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-d844f1603d", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 175", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{OM}{OQ} = \\dfrac{OP}{ON}", "name": null, "statement": "When two chords intersect in a circle, the ratio of two corresponding segments equals the reciprocal of the ratio of the other two; the segments are reciprocally proportional.", "kind": "result", "symbols": [ { "unit": null, "symbol": "OM", "meaning": "segment of chord MN from the intersection point O to M" }, { "unit": null, "symbol": "ON", "meaning": "segment of chord MN from O to N" }, { "unit": null, "symbol": "OP", "meaning": "segment of chord PQ from O to P" }, { "unit": null, "symbol": "OQ", "meaning": "segment of chord PQ from O to Q" } ], "sympy": "Eq(OM/OQ, OP/ON)", "physics": false, "states": [], "concepts": [ "concept/chord", "concept/circle", "concept/proportion", "concept/reciprocal", "concept/similar-polygons" ] }, { "id": "wentworth-plane-geometry-1899/eq-990f5b2b71", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 176", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "AC : AD = AD : AB", "name": null, "statement": "A tangent from an external point is the mean proportional between the whole secant and its external segment.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "the whole secant from A to the far point of the circle" }, { "unit": null, "symbol": "AD", "meaning": "the tangent from A to the circle" }, { "unit": null, "symbol": "AB", "meaning": "the external segment of the secant" } ], "sympy": "Eq(AC/AD, AD/AB)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/geometrical-mean", "concept/proportion", "concept/secant", "concept/tangent" ] }, { "id": "wentworth-plane-geometry-1899/eq-d0aa994786", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 176", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "AC × AB = \\overline{AD}^2", "name": null, "statement": "From a fixed point outside a circle, the product of a secant and its external segment equals the square of the tangent, so it is constant for every secant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AC", "meaning": "whole secant from A" }, { "unit": null, "symbol": "AB", "meaning": "external segment of the secant" }, { "unit": null, "symbol": "AD", "meaning": "tangent from A to the circle" } ], "sympy": "Eq(AC*AB, AD**2)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/product", "concept/secant", "concept/tangent" ] }, { "id": "wentworth-plane-geometry-1899/eq-ccf7423350", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 177", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\overline{NO}^2 = NM × NP - OM × OP", "name": null, "statement": "The square of the bisector of an angle of a triangle equals the product of the two sides enclosing the angle minus the product of the segments the bisector makes on the third side.", "kind": "result", "symbols": [ { "unit": null, "symbol": "NO", "meaning": "the bisector of angle MNP, from N to the opposite side" }, { "unit": null, "symbol": "NM", "meaning": "side of triangle MNP" }, { "unit": null, "symbol": "NP", "meaning": "side of triangle MNP" }, { "unit": null, "symbol": "OM", "meaning": "segment of the third side cut by the bisector, at M" }, { "unit": null, "symbol": "OP", "meaning": "segment of the third side cut by the bisector, at P" } ], "sympy": "Eq(NO**2, NM*NP - OM*OP)", "physics": false, "states": [], "concepts": [ "concept/angle-bisector", "concept/product", "concept/square", "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-1b521387b5", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 181", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{AH}{AC} = \\dfrac{HK}{CE} = \\dfrac{KB}{EF}", "name": null, "statement": "If two lines are cut by any number of parallels, the corresponding intercepts are proportional.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AH", "meaning": "intercept on the first line, from A to H" }, { "unit": null, "symbol": "AC", "meaning": "intercept on the first line, from A to C" }, { "unit": null, "symbol": "HK", "meaning": "intercept between parallels H and K" }, { "unit": null, "symbol": "CE", "meaning": "intercept between parallels C and E" }, { "unit": null, "symbol": "KB", "meaning": "intercept from K to B" }, { "unit": null, "symbol": "EF", "meaning": "intercept from E to F" } ], "sympy": "Eq(AH/AC, KB/EF)", "physics": false, "states": [], "concepts": [ "concept/line-segment", "concept/parallel-lines", "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/eq-d30f217eec", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 181", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "\\dfrac{AH}{m} = \\dfrac{HK}{n} = \\dfrac{KB}{p}", "name": null, "statement": "The construction divides AB into parts AH, HK, KB proportional to the given lines m, n, p.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "given line" }, { "unit": null, "symbol": "n", "meaning": "given line" }, { "unit": null, "symbol": "p", "meaning": "given line" }, { "unit": null, "symbol": "AH", "meaning": "first part of AB produced by the construction" }, { "unit": null, "symbol": "HK", "meaning": "second part of AB" }, { "unit": null, "symbol": "KB", "meaning": "third part of AB" } ], "sympy": "Eq(AH/m, KB/p)", "physics": false, "states": [], "concepts": [ "concept/construction", "concept/line-segment", "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/eq-d3ca1ab9b7", "chapter": "wentworth-plane-geometry-1899/ch-iii", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 185", "location": "PROPORTION\\@. SIMILAR POLYGONS", "latex": "AG:AB = AB:AF", "name": null, "statement": "In dividing a line in extreme and mean ratio, the whole line is the mean proportional between the segments on the chord construction.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "the given line" }, { "unit": null, "symbol": "AG", "meaning": "the longer radius-chord segment from A (AG in the construction)" }, { "unit": null, "symbol": "AF", "meaning": "the shorter radius-chord segment from A (AF in the construction)" } ], "sympy": "Eq(AG/AB, AB/AF)", "physics": false, "states": [], "concepts": [ "concept/construction", "concept/extreme-and-mean-ratio", "concept/geometrical-mean", "concept/proportion" ] }, { "id": "wentworth-plane-geometry-1899/eq-6dd9773ac6", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 194", "location": "AREAS OF POLYGONS", "latex": "\\dfrac{\\rect AF}{\\rect AC} = \\dfrac{AE}{AB}", "name": null, "statement": "Two rectangles on the same altitude are to each other as their bases, so the ratio of the rectangles equals the ratio of the bases AE and AB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AF", "meaning": "rectangle with base AE on the same altitude AD (written \\rect AF)" }, { "unit": null, "symbol": "AC", "meaning": "rectangle with base AB on the same altitude AD (written \\rect AC)" }, { "unit": null, "symbol": "AB", "meaning": "base of rectangle AC" }, { "unit": null, "symbol": "AE", "meaning": "base of rectangle AF" } ], "sympy": "Eq(rect_AF/rect_AC, AE/AB)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-triangle", "concept/common-ratio", "concept/limit", "concept/rectangle", "theorem/area-of-rectangles-with-equal-altitudes" ] }, { "id": "wentworth-plane-geometry-1899/eq-f10a3c8214", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 196", "location": "AREAS OF POLYGONS", "latex": "\\dfrac{R}{U} = \\dfrac{a × b}{1 × 1}", "name": null, "statement": "The ratio of a rectangle R to the unit of surface U equals the product of its altitude and base, each measured in linear units.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "the rectangle" }, { "unit": "unit of surface", "symbol": "U", "meaning": "the unit of surface (a square whose side is the unit of length)" }, { "unit": null, "symbol": "a", "meaning": "altitude of the rectangle R" }, { "unit": null, "symbol": "b", "meaning": "base of the rectangle R" } ], "sympy": "Eq(R/U, a*b/(1*1))", "physics": false, "states": [], "concepts": [ "concept/rectangle", "quantity/area", "theorem/area-of-a-rectangle", "unit/unit", "unit/unit-of-surface" ] }, { "id": "wentworth-plane-geometry-1899/eq-5a61a291b1", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 197", "location": "AREAS OF POLYGONS", "latex": "\\Par AEFD = a × b", "name": null, "statement": "The area of parallelogram AEFD equals its base b times its altitude a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "altitude of the parallelogram" }, { "unit": null, "symbol": "b", "meaning": "base of the parallelogram" } ], "sympy": "Eq(Par_AEFD, a*b)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-triangle", "concept/parallelogram", "quantity/area", "theorem/area-of-a-parallelogram" ] }, { "id": "wentworth-plane-geometry-1899/eq-fe99585815", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 198", "location": "AREAS OF POLYGONS", "latex": "\\triangle{}ABC=\\frac{1}{2}a × b", "name": null, "statement": "The area of a triangle is half the product of its base by its altitude.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "altitude of the triangle ABC" }, { "unit": null, "symbol": "b", "meaning": "base of the triangle ABC" } ], "sympy": "Eq(T_ABC, a*b/2)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-triangle", "concept/base-of-a-triangle", "concept/parallelogram", "concept/triangle", "quantity/area", "theorem/area-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-109140c2f8", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 199", "location": "AREAS OF POLYGONS", "latex": "ABCH=\\frac{1}{2}a(b+b')", "name": null, "statement": "The area of trapezoid ABCH equals half its altitude times the sum of its two bases.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "altitude of the trapezoid" }, { "unit": null, "symbol": "b", "meaning": "one base of the trapezoid" }, { "unit": null, "symbol": "b'", "meaning": "the other base of the trapezoid (written b' in the book; sympy name bp)" } ], "sympy": "Eq(ABCH, a*(b + bp)/2)", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-triangle", "concept/base-of-a-triangle", "concept/trapezoid", "quantity/area", "theorem/area-of-a-trapezoid" ] }, { "id": "wentworth-plane-geometry-1899/eq-6a08926d90", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 200", "location": "AREAS OF POLYGONS", "latex": "\\dfrac{\\triangle ABC} {\\triangle ADE} = \\dfrac{AB × AC} {AD × AE}", "name": null, "statement": "Two triangles with one equal angle at A are to each other as the products of the two sides enclosing that angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ABC", "meaning": "triangle ABC (its area)" }, { "unit": null, "symbol": "ADE", "meaning": "triangle ADE (its area), sharing angle A with ABC" }, { "unit": null, "symbol": "AB", "meaning": "side of triangle ABC from A" }, { "unit": null, "symbol": "AC", "meaning": "side of triangle ABC from A" }, { "unit": null, "symbol": "AD", "meaning": "side of triangle ADE from A" }, { "unit": null, "symbol": "AE", "meaning": "side of triangle ADE from A" } ], "sympy": "Eq(T_ABC/T_ADE, AB*AC/(AD*AE))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/plane-angle", "concept/triangle", "quantity/angle", "theorem/area-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-f873cfefbf", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 202", "location": "AREAS OF POLYGONS", "latex": "S:S'=\\overline{AB}^2:\\overline{A'B'^2}", "name": null, "statement": "The areas of two similar polygons are to each other as the squares of any two homologous sides. (The book's typesetting places the square exponent inside the second overline; the intended relation is the squares of AB and A'B'.)", "kind": "result", "symbols": [ { "unit": "unit of surface", "symbol": "S", "meaning": "area of the first of the two similar polygons" }, { "unit": "unit of surface", "symbol": "S'", "meaning": "area of the second of the two similar polygons" }, { "unit": "unit", "symbol": "AB", "meaning": "a side of the first polygon, homologous to A'B'" }, { "unit": "unit", "symbol": "A'B'", "meaning": "the homologous side of the second polygon" } ], "sympy": "Eq(S/Sp, AB**2/ApBp**2)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/homologous-sides", "concept/similar-polygons", "concept/square", "quantity/area", "theorem/areas-of-similar-polygons" ] }, { "id": "wentworth-plane-geometry-1899/eq-028c911437", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 203", "location": "AREAS OF POLYGONS", "latex": "BE \\Bumpeq CH + AF", "name": "Pythagorean theorem (area form)", "statement": "The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the two legs.", "kind": "law", "symbols": [ { "unit": "unit of surface", "symbol": "BE", "meaning": "square on the hypotenuse BC of right triangle ABC (its area)" }, { "unit": "unit of surface", "symbol": "CH", "meaning": "square on leg AC" }, { "unit": "unit of surface", "symbol": "AF", "meaning": "square on leg AB" } ], "sympy": "Eq(BE, CH + AF)", "physics": false, "states": [ "law/pythagorean-theorem-area-form" ], "concepts": [ "concept/equivalent-plane-figures", "concept/hypotenuse", "concept/leg-of-a-right-triangle", "concept/right-triangle", "concept/square", "quantity/area", "theorem/pythagorean-theorem", "theorem/pythagorean-theorem" ] }, { "id": "wentworth-plane-geometry-1899/eq-66a8987c79", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 210", "location": "AREAS OF POLYGONS", "latex": "\\overline{NP}^2 = MN × NO = a × b", "name": null, "statement": "The square constructed on NP equals the product of the segments MN (= a) and NO (= b) of the diameter, so the square is equivalent to the parallelogram of base b and altitude a.", "kind": "result", "symbols": [ { "unit": "unit", "symbol": "NP", "meaning": "perpendicular at N to the diameter MO; side of the required square" }, { "unit": "unit", "symbol": "MN", "meaning": "segment of the diameter, equal to the altitude a of the parallelogram" }, { "unit": "unit", "symbol": "NO", "meaning": "segment of the diameter, equal to the base b of the parallelogram" }, { "unit": null, "symbol": "a", "meaning": "altitude of the given parallelogram" }, { "unit": null, "symbol": "b", "meaning": "base of the given parallelogram" } ], "sympy": "Eq(NP**2, MN*NO)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/diameter", "concept/equivalent-plane-figures", "concept/geometrical-mean", "concept/parallelogram", "concept/square" ] }, { "id": "wentworth-plane-geometry-1899/eq-868646ac61", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 205", "location": "AREAS OF POLYGONS", "latex": "\\overline{BD}^2 + \\overline{AC}^2 = \\overline{AB}^2 + \\overline{DC}^2", "name": null, "statement": "For a right triangle ABC with right angle at C and a line BD cutting AC at D, the sum of the squares on BD and AC equals the sum of the squares on AB and DC.", "kind": "result", "symbols": [ { "unit": "unit", "symbol": "BD", "meaning": "line from B to D on side AC" }, { "unit": "unit", "symbol": "AC", "meaning": "leg of the right triangle" }, { "unit": "unit", "symbol": "AB", "meaning": "hypotenuse of the right triangle" }, { "unit": "unit", "symbol": "DC", "meaning": "segment of AC from D to C" } ], "sympy": "Eq(BD**2 + AC**2, AB**2 + DC**2)", "physics": false, "states": [], "concepts": [ "concept/hypotenuse", "concept/leg-of-a-right-triangle", "concept/line-segment", "concept/right-triangle", "concept/square", "theorem/pythagorean-theorem" ] }, { "id": "wentworth-plane-geometry-1899/eq-a15819d0c1", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 217", "location": "AREAS OF POLYGONS", "latex": "= \\sqrt{s(s - a)(s - b)(s - c)}", "name": "Heron's formula", "statement": "The area of a triangle with sides a, b, c equals the square root of s(s-a)(s-b)(s-c), where s is half the perimeter.", "kind": "formula", "symbols": [ { "unit": "unit of surface", "symbol": "S", "meaning": "area of the triangle (the left-hand side, given in the book as S)" }, { "unit": "unit", "symbol": "s", "meaning": "semiperimeter, half the sum of the sides (defined earlier in the book; not defined in this chapter excerpt)" }, { "unit": "unit", "symbol": "a", "meaning": "side of the triangle" }, { "unit": "unit", "symbol": "b", "meaning": "side of the triangle" }, { "unit": "unit", "symbol": "c", "meaning": "side of the triangle" } ], "sympy": "Eq(S, sqrt(s*(s - a)*(s - b)*(s - c)))", "physics": false, "states": [ "theorem/heron-s-formula" ], "concepts": [ "concept/side", "concept/triangle", "quantity/perimeter", "theorem/area-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-deae2d0caa", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 217", "location": "AREAS OF POLYGONS", "latex": "= \\dfrac{abc}{4R}", "name": null, "statement": "The area of a triangle equals the product of its three sides divided by four times the radius of its circumscribed circle.", "kind": "formula", "symbols": [ { "unit": "unit of surface", "symbol": "S", "meaning": "area of the triangle (left-hand side in the book)" }, { "unit": "unit", "symbol": "a", "meaning": "side of the triangle" }, { "unit": "unit", "symbol": "b", "meaning": "side of the triangle" }, { "unit": "unit", "symbol": "c", "meaning": "side of the triangle" }, { "unit": "unit", "symbol": "R", "meaning": "radius of the circumscribed circle" } ], "sympy": "Eq(S, a*b*c/(4*R))", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/radius-of-a-regular-polygon", "concept/side", "concept/triangle", "theorem/area-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-34832eefd9", "chapter": "wentworth-plane-geometry-1899/ch-iv", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 217", "location": "AREAS OF POLYGONS", "latex": "\\dfrac{a}{2} × \\dfrac{a\\sqrt{3}}{2} = \\dfrac{a^2\\sqrt{3}}{4}", "name": null, "statement": "The area of an equilateral triangle of side a is a squared times the square root of 3, divided by 4.", "kind": "result", "symbols": [ { "unit": "unit", "symbol": "a", "meaning": "side of the equilateral triangle" }, { "unit": "unit of surface", "symbol": "S", "meaning": "area of the equilateral triangle" }, { "unit": "unit", "symbol": "h", "meaning": "altitude of the equilateral triangle, equal to (a/2)·sqrt(3)" } ], "sympy": "Eq(S, a/2*(a*sqrt(3)/2))", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-triangle", "concept/equilateral-triangle", "concept/root", "quantity/area", "theorem/area-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-893cd7fb48", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 225", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\dfrac{(n-2) 2 \\text{ rt.\\ } \\angle_s}{n}", "name": null, "statement": "Each interior angle of a regular polygon of n sides equals the sum of its interior angles, (n-2) two right angles, divided by n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of sides of the polygon" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/regular-polygon", "concept/sum" ] }, { "id": "wentworth-plane-geometry-1899/eq-4f814b5170", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 227", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\overline{OA}^2 - \\overline{OP}^2 = \\overline{AP}^2", "name": "Pythagorean theorem", "statement": "In the right triangle OAP, the square on the radius OA minus the square on the apothem OP equals the square on AP, half a side.", "kind": "result", "symbols": [ { "unit": null, "symbol": "OA", "meaning": "radius of the circumscribed circle" }, { "unit": null, "symbol": "OP", "meaning": "apothem of the regular inscribed polygon" }, { "unit": null, "symbol": "AP", "meaning": "half of the side AB" } ], "sympy": "Eq(OA**2 - OP**2, AP**2)", "physics": false, "states": [ "theorem/pythagorean-theorem" ], "concepts": [ "concept/apothem", "concept/radius-of-a-regular-polygon", "concept/regular-polygon", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/eq-fca07b2e3a", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 241", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\overline{AD}^2 = DH × DC", "name": null, "statement": "In the right triangle DAH, the square on the side AD equals the product of the hypotenuse DH and its segment DC.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "side of the regular inscribed polygon of double the number of sides" }, { "unit": null, "symbol": "DH", "meaning": "diameter through the centre, equal to 2R" }, { "unit": null, "symbol": "DC", "meaning": "segment of the diameter cut off by the side AB" } ], "sympy": "Eq(AD**2, DH*DC)", "physics": false, "states": [], "concepts": [ "concept/diameter", "concept/regular-polygon", "concept/side", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/eq-292f074c4a", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 241", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "AD = \\sqrt{2-\\sqrt{4-a^2}}", "name": null, "statement": "For a circle of unit radius, the side AD of the polygon with double the sides is found from the side a of the given inscribed polygon.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "side of the regular inscribed polygon of double the number of sides" }, { "unit": null, "symbol": "a", "meaning": "side of the given regular inscribed polygon" } ], "sympy": "Eq(AD, sqrt(2 - sqrt(4 - a**2)))", "physics": false, "states": [], "concepts": [ "concept/radius-of-a-regular-polygon", "concept/regular-polygon", "concept/side" ] }, { "id": "wentworth-plane-geometry-1899/eq-fe10d94775", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 241", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\sqrt{R(2R - \\sqrt{4R^2 - a^2})}", "name": null, "statement": "For a circle of radius R, the side AD of the regular inscribed polygon with double the sides is given in terms of R and the side a of the given polygon.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "side of the regular inscribed polygon of double the number of sides" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle" }, { "unit": null, "symbol": "a", "meaning": "side of the given regular inscribed polygon" } ], "sympy": "Eq(AD, sqrt(R*(2*R - sqrt(4*R**2 - a**2))))", "physics": false, "states": [], "concepts": [ "concept/radius-of-a-regular-polygon", "concept/regular-polygon", "concept/side" ] }, { "id": "wentworth-plane-geometry-1899/eq-29776951fe", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 232", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "S = \\frac{1}{2}R × P", "name": "area of a regular polygon", "statement": "The area of a regular polygon equals half the product of its apothem and its perimeter.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the regular polygon" }, { "unit": null, "symbol": "R", "meaning": "apothem of the regular polygon" }, { "unit": null, "symbol": "P", "meaning": "perimeter of the regular polygon" } ], "sympy": "Eq(S, R*P/2)", "physics": false, "states": [ "theorem/area-of-a-regular-polygon" ], "concepts": [ "concept/apothem", "concept/area", "concept/regular-polygon", "quantity/perimeter" ] }, { "id": "wentworth-plane-geometry-1899/eq-2ac9524257", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 233", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "S' = \\frac{1}{2} R × P", "name": null, "statement": "The area of a regular circumscribed polygon equals half the product of the radius of the circle (its apothem) and the polygon's perimeter, for any number of sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S'", "meaning": "area of the circumscribed regular polygon" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle, the apothem of the circumscribed polygon" }, { "unit": null, "symbol": "P", "meaning": "perimeter of the circumscribed polygon" } ], "sympy": "Eq(S_p, R*P/2)", "physics": false, "states": [], "concepts": [ "concept/apothem", "concept/area", "concept/circumscribed-circle", "concept/regular-polygon", "quantity/perimeter" ] }, { "id": "wentworth-plane-geometry-1899/eq-c03156d208", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 233", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "S = \\frac{1}{2}R× C", "name": null, "statement": "The area of a circle equals half the product of its radius and its circumference.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the circle" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle" }, { "unit": null, "symbol": "C", "meaning": "circumference of the circle" } ], "sympy": "Eq(S, R*C/2)", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle", "concept/radius-of-a-regular-polygon", "quantity/circumference", "theorem/area-of-a-circle" ] }, { "id": "wentworth-plane-geometry-1899/eq-56a60be0c2", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 233", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\odot = \\frac{1}{2} R × C = \\frac{1}{2} R × 2\\pi R = \\pi R^2", "name": null, "statement": "The area of a circle of radius R is pi times the square of its radius.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the circle (written as the circle symbol in the book)" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle" }, { "unit": null, "symbol": "C", "meaning": "circumference of the circle" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(S, pi*R**2)", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle", "concept/radius-of-a-regular-polygon", "quantity/circumference", "quantity/pi", "theorem/area-of-a-circle" ] }, { "id": "wentworth-plane-geometry-1899/eq-0e95c840cf", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 234", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "S:S' = \\pi R^2:\\pi R'^2 = R^2:R'^2", "name": null, "statement": "The areas of two circles are to each other as the squares of their radii.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the first circle" }, { "unit": null, "symbol": "S'", "meaning": "area of the second circle" }, { "unit": null, "symbol": "R", "meaning": "radius of the first circle" }, { "unit": null, "symbol": "R'", "meaning": "radius of the second circle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle", "concept/common-ratio", "concept/radius-of-a-regular-polygon", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-ab83981a82", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 231", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\pi = \\dfrac{C}{2R}", "name": null, "statement": "pi is the constant ratio of the circumference of a circle to its diameter.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "constant ratio of the circumference of a circle to its diameter" }, { "unit": null, "symbol": "C", "meaning": "circumference of the circle" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle" } ], "sympy": "Eq(pi, C/(2*R))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/diameter", "concept/radius-of-a-regular-polygon", "quantity/circumference", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-3ec41d5145", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 231", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "C=2\\pi R", "name": null, "statement": "The circumference of a circle equals 2 pi times its radius.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "circumference of the circle" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(C, 2*pi*R)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/radius-of-a-regular-polygon", "quantity/circumference", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-cef8e23a78", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 242", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "2\\pi R = C", "name": null, "statement": "Twice pi times the radius of a circle equals its circumference.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "circumference of the circle" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle" }, { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(2*pi*R, C)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/radius-of-a-regular-polygon", "quantity/circumference", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-bb8d819cb3", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 242", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\pi = \\frac{1}{2}C", "name": null, "statement": "When the radius is unity, pi equals half the circumference.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter" }, { "unit": null, "symbol": "C", "meaning": "circumference of the unit circle" } ], "sympy": "Eq(pi, C/2)", "physics": false, "states": [], "concepts": [ "concept/radius-of-a-regular-polygon", "quantity/circumference", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-7df4945cbf", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 242", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "C = 6.28317", "name": null, "statement": "The circumference of the unit circle, found by the inscribed-polygon computation, is approximately 6.28317.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "C", "meaning": "circumference of the unit circle" } ], "sympy": "Eq(C, 6.28317)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/limit", "concept/regular-polygon", "quantity/circumference" ] }, { "id": "wentworth-plane-geometry-1899/eq-8dd5758295", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 242", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\pi = 3.14159", "name": null, "statement": "From the computed circumference, pi is nearly 3.14159.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi, 3.14159)", "physics": false, "states": [], "concepts": [ "concept/approximation", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-19b4e5b2d1", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 242", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\pi = 3.1416", "name": null, "statement": "For general use, pi is taken as 3.1416.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(pi, 3.1416)", "physics": false, "states": [], "concepts": [ "concept/approximation", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-f9f6dce470", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 242", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\frac{1}{\\pi} = 0.31831", "name": null, "statement": "For general use, the reciprocal of pi is taken as 0.31831.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "ratio of circumference to diameter" } ], "sympy": "Eq(1/pi, 0.31831)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/numerical-value", "quantity/pi" ] }, { "id": "wentworth-plane-geometry-1899/eq-90091e8382", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 226", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "P:P' = OA:O'A' = OM:O'M'", "name": null, "statement": "The perimeters of two similar regular polygons with the same number of sides are to each other as their circumscribed radii and as their inscribed radii (apothems).", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "perimeter of the first regular polygon" }, { "unit": null, "symbol": "P'", "meaning": "perimeter of the second regular polygon" }, { "unit": null, "symbol": "OA", "meaning": "radius of the first polygon's circumscribed circle" }, { "unit": null, "symbol": "O'A'", "meaning": "radius of the second polygon's circumscribed circle" }, { "unit": null, "symbol": "OM", "meaning": "apothem of the first polygon" }, { "unit": null, "symbol": "O'M'", "meaning": "apothem of the second polygon" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/apothem", "concept/common-ratio", "concept/radius-of-a-regular-polygon", "concept/regular-polygon", "concept/similar", "quantity/perimeter" ] }, { "id": "wentworth-plane-geometry-1899/eq-e39fd6b4df", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 225", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "AB:A'B' = BC:B'C'", "name": null, "statement": "Two regular polygons with the same number of sides have their homologous sides proportional.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "a side of the first regular polygon" }, { "unit": null, "symbol": "A'B'", "meaning": "the homologous side of the second regular polygon" }, { "unit": null, "symbol": "BC", "meaning": "the next side of the first polygon" }, { "unit": null, "symbol": "B'C'", "meaning": "the homologous next side of the second polygon" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/homologous-sides", "concept/regular-polygon", "concept/similar" ] }, { "id": "wentworth-plane-geometry-1899/eq-989b8e4898", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 244", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "\\triangle ACB > \\triangle ADB", "name": null, "statement": "Of two isoperimetric triangles with the same base AB, the isosceles triangle (AC equal to CB) has the greater area.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\triangle ACB", "meaning": "the isosceles triangle ACB (its area)" }, { "unit": null, "symbol": "\\triangle ADB", "meaning": "the other triangle ADB with the same perimeter (its area)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/base-of-a-triangle", "concept/isoperimetric-polygon", "concept/isosceles-triangle", "concept/maximum", "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/eq-a7b5b0e974", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 246", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "ABCDE > A'B'C'D'E'", "name": null, "statement": "A polygon inscribed in a circle has greater area than an equilateral polygon with the same sides that cannot be inscribed in a circle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ABCDE", "meaning": "the inscribed polygon ABCDE (its area)" }, { "unit": null, "symbol": "A'B'C'D'E'", "meaning": "the equilateral polygon A'B'C'D'E' that cannot be inscribed in a circle (its area)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle", "concept/equilateral-polygon", "concept/maximum", "concept/regular-polygon" ] }, { "id": "wentworth-plane-geometry-1899/eq-4e52308087", "chapter": "wentworth-plane-geometry-1899/ch-v", "book": "wentworth-plane-geometry-1899", "edition": "Ginn & Company, revised ed., copyright 1899 (first entered 1888)", "page": "scan 247", "location": "REGULAR POLYGONS AND CIRCLES", "latex": "AB = BC", "name": null, "statement": "In the maximum of isoperimetric polygons with a given number of sides, the adjacent sides AB and BC are equal, because the triangle ABC on AC must be isosceles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "the side AB of the polygon (its length)" }, { "unit": null, "symbol": "BC", "meaning": "the side BC of the polygon (its length)" } ], "sympy": "Eq(AB, BC)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/equilateral-polygon", "concept/isoperimetric-polygon", "concept/isosceles-triangle", "concept/maximum", "concept/side" ] } ], "exercise_sets": [ { "id": "wentworth-plane-geometry-1899/ex-i", "set": "I", "page": "074", "chapter": "wentworth-plane-geometry-1899/ch-i", "practices": [ "concept/altitude-of-a-triangle", "concept/concurrent-lines", "concept/congruent-figures", "concept/diagonal", "concept/equilateral-triangle", "concept/exterior-angle", "concept/isosceles-trapezoid", "concept/isosceles-triangle", "concept/median-of-a-triangle", "concept/parallelogram", "concept/perpendicular-bisector", "concept/rectangle", "concept/rhombus", "concept/right-triangle", "concept/square", "concept/trapezoid", "concept/vertical-angles", "theorem/sum-of-the-angles-of-a-triangle" ] }, { "id": "wentworth-plane-geometry-1899/ex-v-1", "set": "V.1", "page": "250", "chapter": "wentworth-plane-geometry-1899/ch-v", "practices": [ "concept/angle-at-the-centre-of-a-regular-polygon", "concept/apothem", "concept/area", "concept/centre-of-a-circle", "concept/chord", "concept/circle", "concept/circular-sector", "concept/circular-segment", "concept/decagon", "concept/diagonal", "concept/dodecagon", "concept/equiangular-polygon", "concept/equilateral-polygon", "concept/equilateral-triangle", "concept/extreme-and-mean-ratio", "concept/geometrical-mean", "concept/hexagon", "concept/hypotenuse", "concept/inscribed-circle", "concept/isosceles-triangle", "concept/octagon", "concept/pentadecagon", "concept/pentagon", "concept/perimeter", "concept/radius-of-a-regular-polygon", "concept/regular-polygon", "concept/right-triangle", "concept/semicircle", "concept/side", "concept/square", "quantity/arc-of-a-circle" ] }, { "id": "wentworth-plane-geometry-1899/ex-ii-1", "set": "II.1", "page": "117", "chapter": "wentworth-plane-geometry-1899/ch-ii", "practices": [ "concept/altitude-of-a-triangle", "concept/chord", "concept/circle", "concept/circumscribed-circle", "concept/concentric-circles", "concept/construction", "concept/diameter", "concept/equilateral-triangle", "concept/inscribed-angle", "concept/inscribed-circle", "concept/isosceles-trapezoid", "concept/isosceles-triangle", "concept/locus", "concept/parallel-lines", "concept/parallelogram", "concept/perimeter", "concept/perpendicular", "concept/perpendicular-bisector", "concept/point-of-tangency", "concept/quadrilateral", "concept/rectangle", "concept/rhombus", "concept/right-triangle", "concept/secant", "concept/square", "concept/supplementary-angles", "concept/tangent", "concept/tangent-circles", "concept/trapezoid", "quantity/right-angle" ] }, { "id": "wentworth-plane-geometry-1899/ex-ii-2", "set": "II.2", "page": "138", "chapter": "wentworth-plane-geometry-1899/ch-ii", "practices": [ "concept/altitude-of-a-triangle", "concept/angle-bisector", "concept/base-of-a-triangle", "concept/centre-of-a-circle", "concept/circle", "concept/circular-sector", "concept/circumscribed-circle", "concept/construction", "concept/diagonal", "concept/equilateral-triangle", "concept/hypotenuse", "concept/inscribed-circle", "concept/isosceles-trapezoid", "concept/isosceles-triangle", "concept/locus", "concept/median-of-a-triangle", "concept/parallel-lines", "concept/parallelogram", "concept/perimeter", "concept/perpendicular-bisector", "concept/rectangle", "concept/rhombus", "concept/right-triangle", "concept/square", "concept/symmetry", "concept/tangent", "concept/trapezoid", "quantity/distance", "quantity/radius" ] }, { "id": "wentworth-plane-geometry-1899/ex-iv-1", "set": "IV.1", "page": "204", "chapter": "wentworth-plane-geometry-1899/ch-iv", "practices": [] }, { "id": "wentworth-plane-geometry-1899/ex-iv-2", "set": "IV.2", "page": "216", "chapter": "wentworth-plane-geometry-1899/ch-iv", "practices": [] }, { "id": "wentworth-plane-geometry-1899/ex-misc", "set": "Misc", "page": "255", "chapter": "wentworth-plane-geometry-1899/ch-v", "practices": [ "concept/altitude-of-a-triangle", "concept/area", "concept/base-of-a-triangle", "concept/centre-of-a-circle", "concept/chord", "concept/circle", "concept/common-ratio", "concept/diagonal", "concept/equivalent-plane-figures", "concept/extreme-and-mean-ratio", "concept/geometrical-mean", "concept/inscribed-circle", "concept/isosceles-triangle", "concept/line-segment", "concept/locus", "concept/maximum", "concept/minimum", "concept/parallel-lines", "concept/parallelogram", "concept/perimeter", "concept/perpendicular", "concept/point", "concept/point-of-tangency", "concept/semicircle", "concept/square", "concept/tangent", "concept/tangent-circles", "concept/trapezoid", "concept/triangle" ] }, { "id": "wentworth-plane-geometry-1899/ex-iii-1", "set": "III.1", "page": "168", "chapter": "wentworth-plane-geometry-1899/ch-iii", "practices": [] }, { "id": "wentworth-plane-geometry-1899/ex-iii-2", "set": "III.2", "page": "179", "chapter": "wentworth-plane-geometry-1899/ch-iii", "practices": [] }, { "id": "wentworth-plane-geometry-1899/ex-iii-3", "set": "III.3", "page": "187", "chapter": "wentworth-plane-geometry-1899/ch-iii", "practices": [] } ], "problems": [], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }