HMMT 2011 T12

Let $A B C$ 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 $A E F$ lies on the incircle of triangle $A B C$ if and only if the incenter of triangle $A B C$ lies on the circumcircle of triangle $A E F$.