Book 1 - Proposition 1 To construct an equilateral triangle on a given finite straight line. Let [line AB color=fb4934] be the given finite straight line. It is required to construct an equilateral triangle on the straight [line AB]. [step] Describe the [circle BCD radius=AB color=508b0a] with [center A circle=BCD] and [line AB text="radius AB"]. [step] Again describe the [circle ACE radius=AB color=268bd2] with [center B circle=ACE] and [line AB text="radius BA"]. [step] Join the straight lines [line AC color=5f495f] and [line BC color=d33682] from the [point C] at which the circles cut one another to the points [point A] and [point B]. [step] Now, since the [point A] is the center of the [circle BCD], therefore [line AC] equals [line AB]. Again, since the [point B] is the center of the [circle ACE], therefore [line BC] equals [line AB]. But [line AC] was proved equal to [line AB], therefore each of the straight lines [line AC] and [line BC] equals [line AB]. And things which equal the same thing also equal one another, therefore [line AC] also equals [line BC]. Therefore the three straight lines [line AC], [line AB], and [line BC] equal one another. Therefore the [polygon ABC text="triangle ABC"] is equilateral, and it has been constructed on the given finite straight [line AB]. [clear] [polygon ABC hidden] [loc A x=-0.25 y=0] [loc B x=0.25 y=0]