1997 CMO 第 2 题

Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$ .

Consider the sequence of quadrilaterals $A_iB_iC_iD_i$ .

i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?

ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?

设 $A_1B_1C_1D_1$ 为任意凸四边形。 $P$ 是四边形内部的一点，使得由一条边和一条从该边上的一个顶点开始并穿过点 $P$ 的射线所围成的每个角都是锐角。对于 $k\ge 2$ ，我们分别相对于线 $A_{k-1}B_{k-1}、B_{k-1}C_{k-1}、C_{k-1}D_{k-1}、D_{k-1}A_{k-1}$ 递归定义与 $P$ 对称的点 $A_k,B_k,C_k,D_k$ 。

考虑四边形序列 $A_iB_iC_iD_i$ 。

i) 前 12 个四边形中，哪些与 1997 个四边形相似，哪些不相似？

ii) 假设第 1997 个四边形是循环四边形。前 12 个四边形中，哪些是循环四边形，哪些是非循环四边形？