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A Special Cyclic Polygon

Source: 2023 China Team Selection Test Day 1 Problem 1

March 14, 2023
geometryChina TST

Problem Statement

Given an integer n⩾2n \geqslant 2. Suppose there is a point PP inside a convex cyclic 2n2n-gon A1…A2nA_1 \ldots A_{2n} satisfying ∠PA1A2=∠PA2A3=…=∠PA2nA1,\angle PA_1A_2 = \angle PA_2A_3 = \ldots = \angle PA_{2n}A_1,prove that ∏i=1n∣A2i−1A2i∣=∏i=1n∣A2iA2i+1∣, \prod_{i=1}^{n} \left|A_{2i - 1}A_{2i} \right| = \prod_{i=1}^{n} \left|A_{2i}A_{2i+1} \right|,where A2n+1=A1A_{2n + 1} = A_1.
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