TR-MO 2018 Problem 2

Let $P$ be a point in the interior of the triangle $ABC$ . The lines $AP$ , $BP$ , and $CP$ intersect the sides $BC$ , $CA$ , and $AB$ at $D,E$ , and $F$ , respectively. A point $Q$ is taken on the ray $[BE$ such that $E\in [BQ]$ and $m(\widehat{EDQ})=m(\widehat{BDF})$ . If $BE$ and $AD$ are perpendicular, and $|DQ|=2|BD|$ , prove that $m(\widehat{FDE})=60^\circ$ .