Question 82
Let $ABC$ be a triangle and $E, F$ two arbitrary points on the sides $(AB)$ and $(AC)$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ a second time at the point $M$. Let $D$ be the reflection of $M$ across the line $EF$ and $O$ the center of the circumcircle of triangle $ABC$. Prove that $D \in BC$ if and only if $O$ lies on the circumcircle of triangle $AEF$.