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Given a triangle $A B C$. Find the set of centers of rectangles $P Q R S$, where vertices $Q$ and $P$ lie on side $A C$, and vertices $R$ and $S$ lie on sides $A B$ and $B C$ respectively.
Let $O$ - the midpoint of the height $B H, M$ - the midpoint of the segment $A C, D$ and $E$ - the midpoints of sides $R Q$ and $P S$ respectively (fig.).

Points $D$ and $E$ lie on lines $A O$ and $C O$ respectively. The midpoint of segment $D E$ is the center of rectangle $P Q R S$. It is clear that it lies on segment $O M$. The desired locus of points is the segment $O M$, except for its endpoints,

![](https://cdn.mathpix.com/cropped/2024_05_06_8c2fdb390ca80df56784g-39.jpg?height=534&width=672&top_left_y=133&top_left_x=699)

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