Regular polygon inscribed in a unit circle

Question

Given a point $P$ on the circumference of a unit circle and the vertices ${A_1},{A_2}, \ldots ,{A_n}$ of an inscribed regular polygon of $n$ side. Prove that $P{A_1}^4 + P{A_2}^4 + \cdots + P{A_n}^4$ is constant, i.e. independent of the position of $P$.

Answer 1

In complex numbers, this is

$$ \sum_i\left|p-a_i\right|^4=\sum_i(p-a_i)(p-a_i)(\bar p-\bar a_i)(\bar p-\bar a_i)\;. $$

The terms of $0^\text{th}$ and $4^\text{th}$ order in $p$ are constant. The terms of odd orders sum to $0$ by symmetry. The terms of $2^\text{nd}$ order in $p$ that contain either $a_i^2$ or $\bar a_i^2$ sum to $0$ by symmetry, and the term with $p\bar pa_i\bar a_i$ is constant.