2003 APMO 第 2 题

Suppose $A B C D$ is a square piece of cardboard with side length $a$. On a plane are two parallel lines $\ell_{1}$ and $\ell_{2}$, which are also $a$ units apart. The square $A B C D$ is placed on the plane so that sides $A B$ and $A D$ intersect $\ell_{1}$ at $E$ and $F$ respectively. Also, sides $C B$ and $C D$ intersect $\ell_{2}$ at $G$ and $H$ respectively. Let the perimeters of $\triangle A E F$ and $\triangle C G H$ be $m_{1}$ and $m_{2}$ respectively. Prove that no matter how the square was placed, $m_{1}+m_{2}$ remains constant.

假设 $A B C D$ 是一块边长为 $a$ 的方形纸板。平面上有两条平行线 $\ell_{1}$ 和 $\ell_{2}$，它们也相距 $a$ 个单位。将正方形 $A B C D$ 放置在平面上，使得边 $A B$ 和 $A D$ 分别与 $\ell_{1}$ 相交于 $E$ 和 $F$。此外，边 $C B$ 和 $C D$ 分别与 $\ell_{2}$ 相交于 $G$ 和 $H$。设$\triangle A E F$和$\triangle C G H$的周长分别为$m_{1}$和$m_{2}$。证明无论正方形如何放置，$m_{1}+m_{2}$ 保持不变。