Question 173
Suppose $n \ge 2$ is a positive integer, and $A_1A_2 \cdots A_{2n}$ is a convex $2n$-gon inscribed in a circle. It is known that there exists a point $P$ inside this $2n$-gon such that
$$
\angle PA_1A_2 = \angle PA_2A_3 = \cdots = \angle PA_{2n-1}A_{2n} = \angle PA_{2n}A_1.
$$
Prove that the following equation holds
$$
\prod_{i=1}^{n} |A_{2i-1}A_{2i}| = \prod_{i=1}^{n} |A_{2i}A_{2i+1}|,
$$
where $A_{2n+1} = A_1$.
$$
\angle PA_1A_2 = \angle PA_2A_3 = \cdots = \angle PA_{2n-1}A_{2n} = \angle PA_{2n}A_1.
$$
Prove that the following equation holds
$$
\prod_{i=1}^{n} |A_{2i-1}A_{2i}| = \prod_{i=1}^{n} |A_{2i}A_{2i+1}|,
$$
where $A_{2n+1} = A_1$.