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  "body": "\n## Definition\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] 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|01AAYNSBN6]].\nThe incentre of $ABC$ is the unique point $I$ in the interior of the triangle with\n$$d(I, BC) = d(I, CA) = d(I, AB).$$\nThis common distance $r$ is the inradius of $ABC$.\n\n## Remarks\n\nThe point $I$ lies on each of the three internal angle bisectors of the triangle. Thus the three bisectors meet in one point.\nLet $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be the side lengths, with $d$ the [[Euclidean distance|0NPPGHQNXQ]]. Then $I = \\dfrac{aA + bB + cC}{a + b + c}$.\nOutside the triangle there are three more points at the same distance from the three side lines. They are the excentres.\nMathlib's `Affine.Simplex.incenter` defines the incentre of each simplex in a Euclidean affine space.\n\n## Explanations\n\n### L1 human\n\nThe 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 $(1, 1)$.\n",
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