NL-BXMO-TST 2010 Problem 4

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$ . The common tangent that's on the same side as $P$ , intersects the circles at $A$ and $B$ ,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$ , and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$ . Let $E$ be the intersection of $AP$ and $BC$ , and let $F$ be the intersection of $BP$ and $AD$ . Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$ . Prove that $AMBEQF$ is a cyclic hexagon.