CN-TST 2023 Problem 1

Given an integer $n \geqslant 2$ . Suppose there is a point $P$ inside a convex cyclic $2n$ -gon $A_1 \ldots A_{2n}$ satisfying $$\angle PA_1A_2 = \angle PA_2A_3 = \ldots = \angle PA_{2n}A_1,$$ prove that $$\prod_{i=1}^{n} \left|A_{2i - 1}A_{2i} \right| = \prod_{i=1}^{n} \left|A_{2i}A_{2i+1} \right|,$$ where $A_{2n + 1} = A_1$ .