Question 1
In the triangle $ABC$, $AB>AC$, the extension of the altitude $AD$ with $D$ lying inside $BC$ intersects the circumcircle $\omega$ of the triangle $ABC$ at $P$. The circle through $P$ and tangent to $BC$ at $D$ intersects $\omega$ at $Q$ distinct from $P$ with $PQ=DQ$. Prove that $AD=BD-DC$.