USAJMO 2017 Question 3

Let $ABC$ be equilateral and $P$ a point on its circumcircle. Set $D=PA\cap BC$, $E=PB\cap CA$, $F=PC\cap AB$. Prove that $[DEF]=2[ABC]$.

Proposed by Titu Andreescu, Luis Gonzalez, Cosmin Pohoata.