Find the geometric locus of the centers of rectangles inscribed in triangle $ABC$ (one side of the rectangle lies on $AB$).
Let $O$ be the midpoint of the height $C H$, $M$ be the midpoint of the segment $A B$, and $D$ and $E$ be the midpoints of the sides $R Q$ and $P S$ respectively. Points $D$ and $E$ lie on the lines $A O$ and $B O$ respectively. The midpoint of the segment $D E$ is the center of the rectangle $P Q R S$. It is clear that it lies on the segment $O M$. The desired locus of points is the segment $O M$, excluding its endpoints.