ZA-SAMO 2018 Problem 4

Let $ABC$ be a triangle with circumradius $R$ , and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$ . Let $P$ be an arbitrary point in the same plane as $ABC$ . The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation: $$\operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC).$$