OMCC 2011 Problem 2

In a scalene triangle $ABC$ , $D$ is the foot of the altitude through $A$ , $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$ . Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$ , $Y=BE\cap CF$ , $Z=CF \cap AD$ . If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$ , $OYZ$ , $OZX$ must be equilateral.