Question 69
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $M$ be a point in the interior of triangle $ABC$ which is also on the bisector of $\angle A$. Let $AM$, $BM$, $CM$ meet $\Gamma$ in $A_1$, $B_1$, $C_1$ respectively. Suppose $P$ is the point of intersection of $A_1C_1$ with $AB$; and $Q$ is the point of intersection of $A_1B_1$ with $AC$. Prove that $PQ$ is parallel to $BC$.