SHARYGIN 2008 Problem 16

(A.Zaslavsky, 9--11) Given two circles. Their common external tangent is tangent to them at points $A$ and $B$ . Points $X$ , $Y$ on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines $AX$ and $BY$ .