Question 59
In the triangle $ABC$, variable points $D$, $E$, $F$ are on the sides $BC$, $CA$, $AB$ respectively such that the triangle $DFE$ is similar to the triangle $ABC$ in the same order as written. Circumcircles of $BDF$ and $CDE$ intersect the circumcircle of $ABC$ at $P$ and $Q$, respectively, for the second time. Prove that the circumcircle of $DPQ$ passes through a fixed point.