OMM 2016 Problem 6

Let $ABCD$ a quadrilateral inscribed in a circumference, $l_1$ the parallel to $BC$ through $A$ , and $l_2$ the parallel to $AD$ through $B$ . The line $DC$ intersects $l_1$ and $l_2$ at $E$ and $F$ , respectively. The perpendicular to $l_1$ through $A$ intersects $BC$ at $P$ , and the perpendicular to $l_2$ through $B$ cuts $AD$ at $Q$ . Let $\Gamma_1$ and $\Gamma_2$ be the circumferences that pass through the vertex of triangles $ADE$ and $BFC$ , respectively. Prove that $\Gamma_1$ and $\Gamma_2$ are tangent to each other if and only if $DP$ is perpendicular to $CQ$ .