PAMO 2019 Problem 3

Let $ABC$ be a triangle, and $D$ , $E$ , $F$ points on the segments $BC$ , $CA$ , and $AB$ respectively such that $$\frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}.$$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.