CN-TST 2012 Problem 4

Given two circles ${\omega _1},{\omega _2}$ , $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$ , ${\omega _2}$ is the $A$ - excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$ . $S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.