IrMO 2010 Problem 2

Let $ABC$ be a triangle and let $P$ denote the midpoint of the side $BC$ . Suppose that there exist two points $M$ and $N$ interior to the side $AB$ and $AC$ respectively, such that $$|AD|=|DM|=2|DN|,$$ where $D$ is the intersection point of the lines $MN$ and $AP$ . Show that $|AC|=|BC|$ .