USAMO 2009 Problem 5

Trapezoid $ABCD$, with $AB \parallel CD$, is inscribed in circle $\omega$ and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $\omega$ again at points $P$ and $Q$, respectively. Let the line through $G$ parallel to line $AB$ intersect segment $BD$ and $BC$ at points $R$ and $S$, respectively. Prove that quadrilateral $PQRS$ is cyclic if and only if ray $BG$ bisects $\angle CBD$.