2009 USAMO 第 5 题

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

梯形 $ABCD$ 与 $\overline{AB}||\overline{CD}$ 内接于圆 $\omega$，点 $G$ 位于三角形 $BCD$ 内。光线$AG$和$BG$分别在$P$和$Q$处再次与$\omega$相遇。让通过$G$与$\overline{AB}$平行的线分别与$\overline{BD}$和$\overline{BC}$相交于点$R$和$S$。证明四边形$PQRS$是循环的当且仅当$\overline{BG}$平分$\angle CBD$。