Question 14
Let $ABC$ be a triangle, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. If $A$ is not a right angle, prove that the circumcenter of triangle $AEF$ lies on the incircle of triangle $ABC$ if and only if the incenter of triangle $ABC$ lies on the circumcircle of triangle $AEF$.