Problem

Source: INAMO 2019 P3

Tags: Inamo, geometry



Given that $ABCD$ is a rectangle such that $AD > AB$, where $E$ is on $AD$ such that $BE \perp AC$. Let $M$ be the intersection of $AC$ and $BE$. Let the circumcircle of $\triangle ABE$ intersects $AC$ and $BC$ at $N$ and $F$. Moreover, let the circumcircle of $\triangle DNE$ intersects $CD$ at $G$. Suppose $FG$ intersects $AB$ at $P$. Prove that $PM = PN$.