Incentre

Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$. For a point $X$ and a line $\ell$, let $d(X, \ell)$ be the distance from the point to the line. The incentre of $ABC$ is the unique point $I$ in the interior of the triangle with $$d(I, BC) = d(I, CA) = d(I, AB).$$ This common distance $r$ is the inradius of $ABC$.

Remarks

The point $I$ lies on each of the three internal angle bisectors of the triangle. Thus the three bisectors meet in one point. Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be the side lengths, with $d$ the Euclidean distance. Then $I = \dfrac{aA + bB + cC}{a + b + c}$. Outside the triangle there are three more points at the same distance from the three side lines. They are the excentres. Mathlib's Affine.Simplex.incenter defines the incentre of each simplex in a Euclidean affine space.

Explanations

L1 human

The incentre is the point inside a triangle that is equally far from the three sides. You find it as the crossing point of two angle bisectors. For the right triangle with the vertices $(0, 0)$, $(3, 0)$ and $(0, 4)$, the incentre is $(1, 1)$.