CWMO 2017 Problem 6

In acute triangle $ABC$ , let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$ . Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$ .