7.27. Given a triangle $A B C$. Find the set of centers of rectangles $P Q R S$, vertices $Q$ and $P$ of which lie on side $A C$, vertices $R$ and $S$ - on sides $A B$ and $B C$ respectively.
7.27. Let $O$ be the midpoint of the altitude $B H$, $M$ be the midpoint of the segment $A C$, and $D$ and $E$ be the midpoints of sides $R Q$ and $P S$ respectively (Fig. 7.2).

Points $D$ and $E$ lie on the lines $A O$ and $C O$ respectively.

![](https://cdn.mathpix.com/cropped/2024_05_21_22b0ba5e8c5a1f49bd81g-180.jpg?height=322&width=390&top_left_y=1476&top_left_x=542)

Fig. 7.2 It is clear that the midpoint of segment $D E$ is the center of rectangle $P Q R S$. It is evident that it lies on the segment $O M$. The desired geometric locus is the segment $O M$, excluding its endpoints.