Incentre

Builds on Triangle Distance from a point to a line Euclidean distance

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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Euclidean distance concept
Line concept
Normal vector concept
Triangle concept
Distance from a point to a line statement
Incentre definition

Tag	`0TWPTQ0AD7`
URI	`m2://0TWPTQ0AD7`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Incentre
Uses	Triangle, Distance from a point to a line, Euclidean distance

Incentre

Definition

Remarks

Reading path

Further reading

What this page needs for W2

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