2005 USAMO 第 3 题

Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way that convex quadrilateral $APCB_1$ is cyclic, $QB_1 \parallel BA$, and $B_1$ and $Q$ lie on opposite sides of line $AC$. Prove that points $B_1, C_1,P$, and $Q$ lie on a circle.

设$ABC$为锐角三角形，$P$和$Q$为边$BC$上的两点。构造点$C_1$，使得凸四边形$APBC_1$是循环的，$QC_1\平行CA$，并且$C_1$和$Q$位于线$AB$的相对两侧。构造点$B_1$，使得凸四边形$APCB_1$是循环的，$QB_1\平行BA$，并且$B_1$和$Q$位于线$AC$的相对两侧。证明点 $B_1、C_1、P$ 和 $Q$ 位于圆上。