1993 APMO 第 1 题

Let $A B C D$ be a quadrilateral such that all sides have equal length and angle $\angle A B C$ is 60 degrees. Let $\ell$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D)$. Let $E$ and $F$ be the points of intersection of $\ell$ with $A B$ and $B C$ respectively. Let $M$ be the point of intersection of $C E$ and $A F$.

Prove that $C A^{2}=C M \times C E$.

令 $A B C D$ 为四边形，且所有边的长度相等且角度 $\angle A B C$ 为 60 度。令 $\ell$ 为穿过 $D$ 且不与四边形相交的直线($D 处除外)$。设$E$和$F$分别为$\ell$与$A B$和$B C$的交点。令$M$ 为$C E$ 和$A F$ 的交点。

证明 $C A^{2}=C M \times C E$。