FKMO 2017 Problem 5

Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$ . Let $AC \cap BD = E$ , and let rays $AB$ and $DC$ meet again at $F$ . Let $LM \cap DE = P$ . Let $Q$ be the foot of the perpendicular from $P$ to $EM$ . If the orthocenter of $\triangle FLM$ is $E$ , prove the following equality. $$\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$$