Given are the intersection points of a certain rectangle's sides with a line, and the length of one of its sides; construct this rectangle. When can this problem be solved and in how many ways, if it is solvable?
Given the 6 distances determined by points $E, F, G$, and $H$, we draw a semicircle over one of them, for example, $GH$; from point $G$, we draw an arc with the given side $AB = CD = a$, which intersects the semicircle at point $K$.

![](https://cdn.mathpix.com/cropped/2024_05_02_bd8bf5e0510e9654c884g-1.jpg?height=298&width=570&top_left_y=279&top_left_x=772)

We connect $K$ to $H$, and from point $G$, we draw a line parallel to $KH$. From points $E$ and $F$, we draw perpendiculars to this line. This gives us points $A, B, C$, and $D$. If we draw an arc with radius $a$ from point $H$, we get a second solution, which is symmetric to the previous one with respect to the given line. The problem can only be solved if $a < GH$. Since we can perform this procedure with each of the 6 distances, we get a total of $2 \cdot 6 = 12$ solutions. If any of the 6 distances is less than $a$, we get 2 fewer solutions.