IGO 2016 A1

Let the circles $\omega$ and $\omega^ \prime$ intersect in $A$ and $B$ . Tangent to circle $\omega$ at $A$ intersects $\omega^ \prime$ in $C$ and tangent to circle $\omega^ \prime$ at $A$ intersects $\omega$ in $D$ . Suppose that $CD$ intersects $\omega$ and $\omega^ \prime$ in $E$ and $F$ , respectively (assume that $E$ is between $F$ and $C$ ). The perpendicular to $AC$ from $E$ intersects $\omega^ \prime$ in point $P$ and perpendicular to $AD$ from $F$ intersects $\omega$ in point $Q$ (The points $A, P$ and $Q$ lie on the same side of the line $CD$ ). Prove that the points $A, P$ and $Q$ are collinear. Proposed by Mahdi Etesami Fard