Question 154
Let $ABCD$ be a quadrilateral with $AB$ parallel to $CD$ and $AB \lt CD$. Lines $AD$ and $BC$ intersect at a point $P$. Point $X \neq C$ on the circumcircle of triangle $ABC$ is such that $PC = PX$. Point $Y \neq D$ on the circumcircle of triangle $ABD$ is such that $PD = PY$. Lines $AX$ and $BY$ intersect at $Q$.
Prove that $PQ$ is parallel to $AB$.
Prove that $PQ$ is parallel to $AB$.