Problem

Source: 2009 USAMO problem 5

Tags: geometry, geometric transformation, homothety, circumcircle, reflection, Inversion, xtimmyGgettingflamed



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$.