MathNet 2019-part-07 — 184 problems (EN–VI) Question 177

Let $ABC$ be a triangle and $P$ a point inside it. Lines $AP$, $BP$, $CP$ meet the opposite sides at $D$, $E$, $F$. Assume that the three quadrilaterals $PDCE$, $PEAF$, $PFBD$ are all bicentric. Prove that triangle $ABC$ is equilateral.

(A quadrilateral is bicentric if it can be inscribed inside a circle, and it also can have a circle inscribed inside it.)