CN-LEAGUE 2016 Problem 2

Let $X,Y$ be two points which lies on the line $BC$ of $\triangle ABC(X,B,C,Y\text{lies in sequence})$ such that $BX\cdot AC=CY\cdot AB$ , $O_1,O_2$ are the circumcenters of $\triangle ACX,\triangle ABY$ , $O_1O_2\cap AB=U,O_1O_2\cap AC=V$ . Prove that $\triangle AUV$ is a isosceles triangle.