2019 OIM Problems/Problem 4

Problem

Let $ABCD$ be a trapezoid with $AB \parallel CD$ and inscribed in the circle $\Gamma$ . Let $P$ and $Q$ be two points on the segment $AB$ ( $A, P, Q, B$ are in that order and are different) such that $AP = QB$ . Let $E$ and $F$ be the second points of intersection of the lines $CP$ and $CQ$ with $\Gamma$ , respectively. The lines $AB$ and $EF$ intersect at $G$ . Prove that the line $DG$ is tangent to $\Gamma$ .