UA-UMO 2018 Problem 2

In acute-angled triangle $ABC$ , $AH$ is an altitude and $AM$ is a median. Points $X$ and $Y$ on lines $AB$ and $AC$ respectively are such that $AX=XC$ and $AY=YB$ . Prove that the midpoint of $XY$ is equidistant from $H$ and $M$ .

Proposed by Danylo Khilko