RO-NMO 2008 Problem 101

Let $ABC$ be a triangle and the points $D\in (BC)$ , $E\in (CA)$ , $F\in (AB)$ such that $$\frac {BD}{DC} = \frac {CE}{EA} = \frac {AF}{FB}.$$ Prove that if the circumcenters of the triangles $DEF$ and $ABC$ coincide then $ABC$ is equilateral.