Question 122
The two circles $\Gamma_{1}$ and $\Gamma_{2}$ intersect in $P$ and $Q$. The common tangent on the $P$ side hits the circles in $A$ resp. $B$. The tangent to $\Gamma_{1}$ in $P$ cuts $\Gamma_{2}$ for the second time in $C$ and the tangent line to $\Gamma_{2}$ $P$ cuts $\Gamma_{1}$ for the second time in $D$. We call the intersection of the lines $A P$ and $B C$ $E$ and the intersection of the lines $B P$ and $A D$ $F$. $M$ $P$ point mirror in the middle of $A B$. Prove that $A M B E Q F$ is a cord hexagon.