Incentre - concurrency of the bisectors

The incentre is the intersection of the three bisectors of the interior angles and it is equidistant from the three sides: the common distance is the radius of the inscribed circle.

Theorem — Concurrency of the bisectors

The three bisectors of the interior angles of a triangle meet at a single point $I$ (incentre), equidistant from the three sides.

Proof — The bisectors are concurrent

1. Let $b_A$ be the bisector of the angle $\widehat{A}$ and $b_B$ that of the angle $\widehat{B}$, and let $I$ be their point of intersection.
2. Since $I$ lies on the bisector of $\widehat{A}$, it is equidistant from the sides $AB$ and $AC$: $d(I,AB)\cong d(I,AC)$.
3. Since $I$ lies on the bisector of $\widehat{B}$, it is equidistant from the sides $AB$ and $BC$: $d(I,AB)\cong d(I,BC)$.
4. By transitivity: $d(I,AC)\cong d(I,BC)$, hence $I$ also lies on the bisector of $\widehat{C}$: the three bisectors are concurrent at $I$.
5. The common distance $r = d(I, \text{lato})$ is the radius of the inscribed circle.

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Links

Topics: Euclidean circle
Concepts: Bisector · Inscribed circle · Incentre · Line
Skills: Proving · Synthetic geometry