Problem

Source: USAMO 2019 Problem 2 and JMO 2019 Problem 3, by Ankan Bhattacharya

Tags: USA(J)MO, USAMO, Hi, xt



Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$. Proposed by Ankan Bhattacharya