Question 115
$ABCD$ is a trapezoid with $AB \parallel CD$. Suppose that 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 $AD$.