KöMaL 2020 B5096

In a regular triangle $ABC$ of unit sides, let $P$ be an arbitrary point on the circumference of the incircle. Let $D$, $E$, and $F$ denote the orthogonal projections of point $P$ on the sides $BC$, $AC$ and $AB$, respectively. Prove that the area of triangle $DEF$ is a constant, independent of the choice of $P$.