USAJMO 2019 Question 3

Let $ABCD$ be a cyclic quadrilateral with $AD^2+BC^2=AB^2$. Its diagonals meet at $E$. A point $P$ on side $AB$ satisfies $\angle APD=\angle BPC$. Prove that line $PE$ passes through the midpoint of $CD$.

Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party solution notes.