2009 USAMO Problems/Problem 5

Problem

Trapezoid $ABCD$ , with $\overline{AB}||\overline{CD}$ , 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 $\overline{AB}$ intersects $\overline{BD}$ and $\overline{BC}$ at points $R$ and $S$ , respectively. Prove that quadrilateral $PQRS$ is cyclic if and only if $\overline{BG}$ bisects $\angle CBD$ .