Question 76
Let $ABC$ be an acute triangle and let $k$ be the circle circumscribed around it. The point $O$ in the interior of the triangle is such that $\overline{CE} = \overline{CF}$, where $E$ and $F$ are points on $k$ and $E$ lies on $AO$, and $F$ lies on $BO$. Prove that $O$ lies on the bisector of the angle at the vertex $C$ if and only if the triangle is isosceles with base $AB$.