MathNet 2006 — Part 02 Question 25

Let $ABC$ be a right triangle in $A$. Consider all triangles $XYZ$, isosceles rectangles in $X$, where $X$ is on segment $BC$, $Y$ is on segment $AB$, and $Z$ is on segment $AC$.
Determine the locus of the midpoints of the hypotenuses $YZ$ of such triangles $XYZ$.