Exercise 20. Three lines $D_{1}, D_{2}, D_{3}$ are parallel to each other. The line $D_{2}$ is located between the other two. It is 13 centimeters from $D_{1}$ and 31 centimeters from $D_{3}$. Determine the area, expressed in square centimeters, of a square in which each of these three lines contains one vertex.
Solution to Exercise 20: Let $A B C D$ be the square such that $A \in D_{1}, B \in D_{2}$ and $C \in D_{3}$. Let $x = A B$. We denote $\alpha$ as the angle between $D_{2}$ and $(B A)$. Then $\sin \alpha = \frac{13}{x}$ and $\cos \alpha = \frac{31}{x}$, so $x^{2} = 13^{2} + 31^{2} = 1130$.