Problem

Source: Mexico National Olympiad 2019 Problem 2

Tags: geometry, orthocenter, perpendicular lines, perpendicular bisector



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