2011 USAMO 第 5 题

Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that $\angle Q_1 BC = \angle ABP$, $\angle Q_1 CB = \angle DCP$, $\angle Q_2 AD = \angle BAP$, $\angle Q_2 DA = \angle CDP$. Prove that $\overline{Q_1 Q_2} \parallel \overline{AB}$ if and only if $\overline{Q_1 Q_2} \parallel \overline{CD}$.

令 $P$ 为四边形 $ABCD$ 内的给定点。点 $Q_1$ 和 $Q_2$ 位于 $ABCD$ 内，使得 $\angle Q_1 BC = \angle ABP$、$\angle Q_1 CB = \angle DCP$、$\angle Q_2 AD = \angle BAP$、$\angle Q_2 DA = \angle CDP$。证明 $\overline{Q_1 Q_2} \parallel \overline{AB}$ 当且仅当 $\overline{Q_1 Q_2} \parallel \overline{CD}$。