Let's construct a rectangle that passes through four given points and has one of its sides given.

Construct a rectangle, whose sides pass through four given points and one of its sides is given.
Let the given points be $A, B, C$, and $D$. Connect $B$ and $D$, draw a semicircle over $BD$, and cut off a distance $a$ from $D$ along the circle, which intersects the circle at $F$. Connect $F$ and $B$, draw perpendiculars from $A$ and $C$ to $FB$, and draw a line parallel to $BF$ from $D$. This way, we obtain the desired rectangle $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_854bda03fc0d01e4ab4fg-1.jpg?height=303&width=290&top_left_y=304&top_left_x=907)

If the distance $a$ is measured from $B$ along the circle, we obtain a rectangle congruent to the previous one. Since this procedure can be applied to every pair of points, the problem generally has 12 solutions. However, if the distance between two points is less than the given distance $a$, we get fewer solutions, specifically 2 fewer solutions.

(Appel Sándor, Déva.)

The problem was also solved by: Boros J., Czank K., Filkorn J., Fleischer F., Freibauer E., Kéler E., Kerekes T., Krausz B., Krisztián Gy., Lukhaub Gy., Lupsa Gy., Obláth R., Sasvári G., Szabó J., Vajda Ö., Weisz J.