OMM 2019 Problem 2

Let $H$ be the orthocenter of acute-angled triangle $ABC$ and $M$ be the midpoint of $AH$ . Line $BH$ cuts $AC$ at $D$ . Consider point $E$ such that $BC$ is the perpendicular bisector of $DE$ . Segments $CM$ and $AE$ intersect at $F$ . Show that $BF$ is perpendicular to $CM$ .

Proposed by Germán Puga