Question 23
Let $ABC$ be a triangle. The excircle $\omega$ opposite the vertex $A$ touches the segment $\overline{BC}$ and the rays $AC$ and $AB$ at the points $D$, $E$ and $F$ respectively. The circumcircle of triangle $AEF$ meets the line $BC$ at the points $P$ and $Q$. Finally, let $M$ be the midpoint of the segment $\overline{AD}$. Prove that $\omega$ and the circumcircle of triangle $MPQ$ are tangent.