IR-TST 2017 Problem 7

$ABCD$ is a trapezoid with $AB \parallel CD$ . The diagonals intersect at $P$ . Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$ . Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$ . $\omega _3$ is the circumcircle of triangle $BPC$ . Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$ .

Proposed by Kasra Ahmadi