1992 IMO 第 5 题

(COL 3) Let $A B C D$ be a convex quadrilateral such that $A C=$ $B D$. Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_{1}, O_{2}, O_{3}, O_{4}$ be the centers of the triangles constructed on $A B, B C, C D, D A$ respectively. Show that $O_{1} O_{3}$ is perpendicular to $O_{2} O_{4}$.

(COL 3) 令$A B C D$ 为凸四边形，使得$A C=$ $B D$。等边三角形是在四边形的边上构造的。设$O_{1}、O_{2}、O_{3}、O_{4}$分别为在$A B、B C、C D、D A$上构造的三角形的中心。证明 $O_{1} O_{3}$ 垂直于 $O_{2} O_{4}$。