RMO 2016 A5

Let $ABC$ be a triangle with centroid $G$ . Let the circumcircle of triangle $AGB$ intersect the line $BC$ in $X$ different from $B$ ; and the circucircle of triangle $AGC$ intersect the line $BC$ in $Y$ different from $C$ . Prove that $G$ is the centroid of triangle $AXY$ .