US-TST 2020 Problem 6

Let $P_{1} P_{2} \ldots P_{100}$ be a cyclic 100-gon, and let $P_{i}=P_{i+100}$ for all $i$. Define $Q_{i}$ as the intersection of diagonals $\overline{P_{i-2} P_{i+1}}$ and $\overline{P_{i-1} P_{i+2}}$ for all integers $i$. Suppose there exists a point $P$ satisfying $\overline{P P_{i}} \perp \overline{P_{i-1} P_{i+1}}$ for all integers $i$. Prove that the points $Q_{1}, Q_{2}, \ldots, Q_{100}$ are concyclic.