SHARYGIN 2007 Problem 13

On the side $AB$ of a triangle $ABC$ , two points $X, Y$ are chosen so that $AX = BY$ . Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$ . Prove that all lines $UV$ (for all $X, Y$ , given $A, B, C$ ) have a common point.