KöMaL 2015 B4749

The feet of the altitudes drawn from vertices $B$ and $C$ of an acute-angled triangle $ABC$ on the sides $AC$ and $AB$ are $D$ and $E$ , respectively. The midpoint of side $BC$ is $F$ . The intersection of line segments $AF$ and $DE$ is $M$ , and the orthogonal projection of point $M$ onto the line segment $BC$ is $N$ . Prove that line segment $AN$ bisects line segment $DE$ .