DE-MO 2011 Problem 3

Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$ . A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$ . The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$ . Prove $BN=LC$ .