Question 162
Let $\triangle ABC$ 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:
$$
\text{area}(DEF) = \frac{PH^2}{4R^2} \text{area}(ABC).
$$
$$
\text{area}(DEF) = \frac{PH^2}{4R^2} \text{area}(ABC).
$$