CONO-SUR 2014 Problem 5

Let $ABCD$ be an inscribed quadrilateral in a circumference with center $O$ such that it lies inside $ABCD$ and $\angle{BAC} = \angle{ODA}$ . Let $E$ be the intersection of $AC$ with $BD$ . Lines $r$ and $s$ are drawn through $E$ such that $r$ is perpendicular to $BC$ , and $s$ is perpendicular to $AD$ . Let $P$ be the intersection of $r$ with $AD$ , and $M$ the intersection of $s$ with $BC$ . Let $N$ be the midpoint of $EO$ .

Prove that $M$ , $N$ , and $P$ lie on a line.