CZ-SK-MO 2014 Problem 2

A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$ , triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$ .

(Day 1, 2nd problem authors: Michal Rolínek, Jaroslav Švrček)