Let $ ABCD$ be a convex quadrtilateral such that $ AB$ is not parallel with $ CD$. Let $ \Gamma_1$ be a circle that passes through $ A$ and $ B$ and is tangent to $ CD$ at $ P$. Also, let $ \Gamma_2$ be a circle that passes through $ C$ and $ D$ and is tangent to $ AB$ at $ Q$. Let the circles $ \Gamma_1$ and $ \Gamma_2$ intersect at $ E$ and $ F$. Prove that $ EF$ passes through the midpoint of $ PQ$ iff $ BC \parallel AD$.