KöMaL 2005 B3839

The bisector of the angle $A$ of triangle ABC intersects the side $BC$ at $D$. The lines $b$ and $c$ drawn through the points $B$ and $C$, respectively, are parallel to each other and equidistant from $A$. Let $M$ and $N$ denote the points of the lines $b$and $c$, such that $AB$ bisects the line segment $DM$ and $AC$ bisects the line segment $DN$. Prove that $DM$=$DN$.