TR-MO 2014 Problem 3

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$ , respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$ . Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$ , respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$ , respectively such that the equation $$\frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN}$$ holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.