--- author: Stéphane Laurent date: '2020-01-16' highlighter: 'pandoc-solarized' linenums: True output: html_document: highlight: kate keep_md: False md_document: preserve_yaml: True variant: markdown prettify: True prettifycss: minimal tags: 'R, maths, geometry' title: Malfatti circles --- The [Malfatti cirlces](https://en.wikipedia.org/wiki/Malfatti_circles) of a triangle can be constructed using [straightedge and compass](https://math.stackexchange.com/a/333522/38217). Here we provide the analytical formulae of the Malfatti circles of a triangle $ABC$. Let $I$ be the incircle of $ABC$ and $r$ its inradius. Let $a = BC$, $b = AC$ and $c = AB$ be the edge lengths of $ABC$, and $s = \frac{1}{2}(a+b+c)$ be the semiperimeter. The radii of the Malfatti circles are given by $$r_1 = \frac{r\bigl(s - r - (IB+IC-IA)\bigr)}{2(s-a)},$$ $$r_2 = \frac{r\bigl(s - r - (IC+IA-IB)\bigr)}{2(s-b)},$$ $$r_3 = \frac{r\bigl(s - r - (IA+IB-IC)\bigr)}{2(s-c)}.$$ Now, the centers. Set $$d_1 = \frac{r_1}{\tan\left(\dfrac{\arccos\left(\dfrac{(C-A).(B-A)}{bc}\right)}{2}\right)},$$ $$d_2 = \frac{r_2}{\tan\left(\dfrac{\arccos\left(\dfrac{(C-B).(A-B)}{ac}\right)}{2}\right)},$$ $$d_3 = \frac{r_3}{\tan\left(\dfrac{\arccos\left(\dfrac{(A-C).(B-A)}{ba}\right)}{2}\right)},$$ $$w = d_1 + d_2 + \sqrt{r_1r_2},$$ $$u = d_2 + d_3 + \sqrt{r_2r_3},$$ $$v = d_3 + d_1 + \sqrt{r_3r_1},$$ $$d = \frac{\sqrt{(-u+v+w)(u+v-w)(u-v+w)(u+v+w)}}{2}.$$ Then the [trilinear coordinates](https://en.wikipedia.org/wiki/Trilinear_coordinates) $x:y:z$ of the center $O_1$ are $$x = \frac{d}{r_1} - (v+w), \quad y = u, \quad z = u.$$ For $O_2$, $$x = v, \quad y = \frac{d}{r_2} - (u+w), \quad z = v.$$ And for $O_3$, $$x = w, \quad y = w, \quad z = \frac{d}{r_3} - (u+v).$$ Here is a R code returning the Malfatti circles:  {.r} MalfattiCircles <- function(A, B, C){ a <- sqrt(c(crossprod(B-C))) # distance BC b <- sqrt(c(crossprod(A-C))) # distance AC c <- sqrt(c(crossprod(B-A))) # distance AB p <- (a + b + c); s <- p / 2; areaABC <- sqrt(s*(s-a)*(s-b)*(s-c)) I <- (A*a + B*b + C*c) / p # incenter r <- areaABC / s # inradius # radii of Malfatti circles #### IA <- sqrt(c(crossprod(I-A))) IB <- sqrt(c(crossprod(I-B))) IC <- sqrt(c(crossprod(I-C))) r1 <- r * (s-r-(IB+IC-IA)) / 2 / (s-a) r2 <- r * (s-r-(IC+IA-IB)) / 2 / (s-b) r3 <- r * (s-r-(IA+IB-IC)) / 2 / (s-c) # centers of Malfatti circles #### d1 <- r1 / tan(acos(c(crossprod(C-A,B-A)/b/c))/2) d2 <- r2 / tan(acos(c(crossprod(C-B,A-B)/a/c))/2) d3 <- r3 / tan(acos(c(crossprod(A-C,B-C)/b/a))/2) w <- d1 + d2 + 2*sqrt(r1*r2) u <- d2 + d3 + 2*sqrt(r2*r3) v <- d3 + d1 + 2*sqrt(r3*r1) d <- sqrt((-u+v+w)*(u+v-w)*(u-v+w)*(u+v+w))/2 x <- d/r1 - (v+w); y <- u; z <- u # trilinear coordinates O1 <- (u*x*A + v*y*B + w*z*C) / (u*x + v*y + w*z) x <- v; y <- d/r2 - (u+w); z <- v # trilinear coordinates O2 <- (u*x*A + v*y*B + w*z*C) / (u*x + v*y + w*z) x <- w; y <- w; z <- d/r3 - (u+v) # trilinear coordinates O3 <- (u*x*A + v*y*B + w*z*C) / (u*x + v*y + w*z) return(list( C1 = list(center = O1, radius = r1), C2 = list(center = O2, radius = r2), C3 = list(center = O3, radius = r3) )) }  Try it:  {.r} A <- c(0, 0) B <- c(1.5, 2) C <- c(2, 0.5) Mcircles <- MalfattiCircles(A, B, C) C1 <- Mcircles[[1]]; C2 <- Mcircles[[2]]; C3 <- Mcircles[[3]] O1 <- C1$center; O2 <- C2$center; O3 <- C3$center r1 <- C1$radius; r2 <- C2$radius; r3 <- C3$radius library(plotrix) par(mar=c(3, 3, 1, 1)) plot(0 , 0, type = "n", xlim = c(0,2), ylim = c(0,2), asp = 1, xlab = NA, ylab = NA) points(rbind(A,B,C), pch = 19, col = c("red", "green", "blue")) lines(rbind(A,B,C,A)) # draw triangle ABC draw.circle(O1[1], O1[2], r1, col = "red") draw.circle(O2[1], O2[2], r2, col = "green") draw.circle(O3[1], O3[2], r3, col = "blue")  ![](figures/Malfatti-plotMalfatti-1.png)