ELMO SL 2014 G3

Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$ -gon. Prove that for every point $P$ not the circumcenter of the $2013$ -gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$ .

Proposed by Robin Park