Question 50
Let $\triangle ABC$ be a triangle and let $\omega$ be its $A$-excircle (excircle opposite vertex $A$). Let $D, E, F$ be the points where $\omega$ touches the lines $BC, CA, AB$, respectively. The circle $AEF$ crosses the line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of the line segment $AD$. Prove that the circles $\omega$ and $MPQ$ are tangent.