CN-SMO 2015 Problem 7

In $\triangle ABC$ , we have $AB>AC>BC$ . $D,E,F$ are the tangent points of the inscribed circle of $\triangle ABC$ with the line segments $AB,BC,AC$ respectively. The points $L,M,N$ are the midpoints of the line segments $DE,EF,FD$ . The straight line $NL$ intersects with ray $AB$ at $P$ , straight line $LM$ intersects ray $BC$ at $Q$ and the straight line $NM$ intersects ray $AC$ at $R$ . Prove that $PA \cdot QB \cdot RC = PD \cdot QE \cdot RF$ .