JBMO-SL 2002 Problem 10

Let $ABC$ be a triangle with area $S$ and points $D,E,F$ on the sides $BC,CA,AB$ . Perpendiculars at points $D,E,F$ to the $BC,CA,AB$ cut circumcircle of the triangle $ABC$ at points $(D_1,D_2), (E_1,E_2), (F_1,F_2)$ . Prove that: $|D_1B\cdot D_1C - D_2B\cdot D_2C| + |E_1A\cdot E_1C - E_2A\cdot E_2C| + |F_1B\cdot F_1A - F_2B\cdot F_2A| > 4S$