Question 34
We are given a segment $AB$ in the plane. Consider a triangle $XYZ$ with the following properties: the vertex $X$ is an interior point of the segment $AB$, the triangles $XBY$ and $XZA$ are similar ($\triangle XBY \sim \triangle XZA$) and the points $A$, $B$, $Y$, $Z$ lie on a circle in this order. Find the locus of midpoints of the sides $YZ$ of all such triangles $XYZ$.