Question 114
Let $ABCD$ be a cyclic quadrilateral. The points $M, N, P$ and $Q$ are the midpoints of the sides $DA, AB, BC$ and $CD$, respectively, and the point $E$ is the intersection of the diagonals $AC$ and $BD$. The circumcircles of $\triangle EMN$ and $\triangle EPQ$ intersect at a point $F \neq E$. Prove that $EF \perp AC$.