Question 40
Let $P_1P_2 \cdots P_{100}$ 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 $\overline{PP_i} \perp \overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1, Q_2, \dots, Q_{100}$ are concyclic.
Suppose there exists a point $P$ satisfying $\overline{PP_i} \perp \overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1, Q_2, \dots, Q_{100}$ are concyclic.