RO-NMO 2008 Problem 71

Let $ABC$ be an acute angled triangle with $\angle B > \angle C$ . Let $D$ be the foot of the altitude from $A$ on $BC$ , and let $E$ be the foot of the perpendicular from $D$ on $AC$ . Let $F$ be a point on the segment $(DE)$ . Show that the lines $AF$ and $BF$ are perpendicular if and only if $EF\cdot DC = BD \cdot DE$ .