USAMO 2005 Problem 3

Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$ . Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is cyclic, $QC_{1}\parallel CA$ , and $C_{1}$ and $Q$ lie on opposite sides of line $AB$ . Construct a point $B_{1}$ in such a way that the convex quadrilateral $APCB_{1}$ is cyclic, $QB_{1}\parallel BA$ , and $B_{1}$ and $Q$ lie on opposite sides of line $AC$ . Prove that the points $B_{1}$ , $C_{1}$ , $P$ , and $Q$ lie on a circle.