Given the acute triangle $A B C$. Determine the geometric locus of the intersection points of the diagonals of rectangles, two of whose vertices lie on the side $A B$, and one vertex each on the sides $A C$ and $B C$.

Determine the geometric locus of the intersection points of the diagonals of rectangles, two of whose vertices lie on the side $A B$, and one vertex each on the sides $A C$ and $B C$.
Let's inscribe a suitable $P_{1} P_{2} P_{3} P_{4}$ rectangle into a triangle, where $P_{1}, P_{2}$ are points on the side $A B$ and $P_{3} P_{4} \| P_{1} P_{2}$. It is known that the intersection of the diagonals coincides with the intersection of the midlines of the rectangle, denoted by $M$.

![](https://cdn.mathpix.com/cropped/2024_05_02_0e5de3872f099368f381g-1.jpg?height=889&width=973&top_left_y=249&top_left_x=562)

The midpoint $Q$ of the side $P_{3} P_{4}$ of the rectangle, which is parallel to $A B$, lies on the median $C D$ from the vertex $C$ of the triangle $A B C$, since this median bisects every segment parallel to $A B$. If we draw a perpendicular from $Q$ to $A B$, the segment $Q R$ extending to $A B$ is one of the midlines of the rectangle, and $M$ is the midpoint of $Q R$. Therefore, it lies on the median from $D$ in the triangle $C D E$, where $E$ is the projection of $C$ onto $A B$. (If $E$ coincides with $D$, i.e., $A C = B C$, then the points $C, D, E$ lie on a straight line, and $M$ is also on this line.) Let the midpoint of $C E$ be $F$, then the desired geometric locus is the segment $D F$.

We will show that for every interior point of the segment $D F$, there exists a corresponding rectangle. Take any point $M$ on the segment $D F$ and denote the intersection of the line through $M$ perpendicular to $A B$ with $C D$ as $Q$, and its intersection with $A B$ as $R$. The line through $Q$ parallel to $A B$ intersects the sides $A C$ and $B C$ at the points $P_{3}$ and $P_{4}$, which are the vertices of the desired rectangle. In the rectangle, $M$ is indeed the intersection of the diagonals, as it is a point on the median $D F$, thus it bisects $Q R$ as well. Furthermore, $Q$ is a point on the median $C D$, which bisects every segment parallel to $A B$, including $P_{3} P_{4}$.