Problem — Centre on the angle bisector

Text

A circle with centre $O$ is tangent to both sides of an angle with vertex $V$. Prove that the centre $O$ lies on the bisector of the angle.

Solution

Let $A$ and $B$ be the points where the circle touches the two sides of the angle. The radii $OA$ and $OB$ are perpendicular to the sides at the points of tangency and both equal $r$. The triangles $OAV$ and $OBV$ are right-angled at $A$ and $B$, share the hypotenuse $OV$, and have legs $\overline{OA} = \overline{OB} = r$: by the congruence criterion for right triangles they are congruent. It follows that $\widehat{OVA} = \widehat{OVB}$, that is, the ray $VO$ divides the angle into two congruent parts: the centre $O$ lies on the bisector. $\blacksquare$