Three excircles (excenters) exist — one interior bisector + two exterior bisectors are concurrent

Dependencies: Angle bisector ⇔ equidistant from sides (including its converse: a point equidistant from the two sides of an angle lies on its bisector), External angle bisector property (a point on the exterior bisector is equidistant from the opposite side and the extension of an adjacent side).

Statement

Let $\triangle ABC$ be a non-degenerate triangle. Fix a vertex (say $A$); then the interior bisector of $\angle A$ and the exterior bisectors of $\angle B$ and $\angle C$ all meet at a single point $I_A$. This point is called the excenter of $\triangle ABC$ opposite vertex $A$. Equivalently, $I_A$ is at equal distance from the three lines $BC$, $AB$, $AC$:

Denote this common distance by $r_A$; then the circle centered at $I_A$ with radius $r_A$ is simultaneously tangent to the interior of side $BC$, to the extension of $AB$ beyond $B$, and to the extension of $AC$ beyond $C$ — this is the excircle $\omega_A$ of $\triangle ABC$. Each of the three vertices yields an excenter, so there are three excenters $I_A$, $I_B$, $I_C$ and three excircles.