2008 IMO Shortlist G3

Let $A B C D$ be a convex quadrilateral and let $P$ and $Q$ be points in $A B C D$ such that $P Q D A$ and $Q P B C$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $P Q$ such that $\angle P A E=\angle Q D E$ and $\angle P B E=\angle Q C E$. Show that the quadrilateral $A B C D$ is cyclic.

令$A B C D$ 为凸四边形，并令$P$ 和$Q$ 为$A B C D$ 中的点，使得$P Q D A$ 和$Q P B C$ 为循环四边形。假设线段$P Q$ 上存在点$E$，使得$\angle P A E=\angle Q D E$ 且$\angle P B E=\angle Q C E$。证明四边形 $A B C D$ 是循环的。