Incenter exists — the three angle bisectors of a triangle meet at one point

Dependencies: Angle bisector ⇔ equidistant from sides (on the line ⇒ equidistant from the two sides), Angle bisector test (equidistant from the two sides ⇒ on the bisector).

Statement

Let $\triangle ABC$ be a non-degenerate triangle. Then the bisectors of the three interior angles $\angle A$, $\angle B$, $\angle C$ meet at a single point $I$; this point is called the incenter of $\triangle ABC$. Equivalently, there is a unique point $I$ inside the triangle such that

Denote this common distance by $r$; then the circle centered at $I$ with radius $r$ is tangent to all three sides (the incircle, treated as a separate theorem in the next section).