2000 IMO Shortlist S14

Let $A_1A_2 \ldots A_n$ be a convex polygon, $n \geq 4.$ Prove that $A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $A_j$ one can assign a pair $(b_j, c_j)$ of real numbers, $j = 1, 2, \ldots, n,$ so that $A_iA_j = b_jc_i - b_ic_j$ for all $i, j$ with $1 \leq i < j \leq n.$

设 $A_1A_2 \ldots A_n$ 为凸多边形， $n \geq 4.$ 证明 $A_1A_2 \ldots A_n$ 是循环的当且仅当每个顶点 $A_j$ 可以分配一对 $(b_j, c_j)$ 实数， $j = 1, 2, \ldots, n,$ 使得 $A_iA_j = b_jc_i - b_ic_j$ 对于所有 $i, j$ 且 $1 \leq i < j \leq n.$