APMO 1998 Problem 4

Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$ . Let $E$ and $F$ lie on a line passing through $D$ such that $AE$ is perpendicular to $BE$ , $AF$ is perpendicular to $CF$ , and $E$ and $F$ are different from $D$ . Let $M$ and $N$ be the midpoints of the segments $BC$ and $EF$ , respectively. Prove that $AN$ is perpendicular to $NM$ .