## Task 3 - 230713

Let $A B C D$ be a rectangle, whose diagonals intersect at point $S$. The angle $\angle A S B$ has a size of $120^{\circ}$.

Determine the lengths of the diagonals $A C$ and $B D$ in dependence on the side length $B C$!
Since in a rectangle the diagonals are of equal length and bisect each other, it follows that

(1) $A S=B S=C S=D S$

Thus, triangle $B C S$ is isosceles, and therefore

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0466.jpg?height=252&width=368&top_left_y=2400&top_left_x=273)

(2) $\angle C B S=\angle B C S$

By the angle sum property of a triangle, we have

(3) $\angle C B S+\angle B C S+60^{\circ}=180^{\circ}$

From (2) and (3), it follows that $\angle C B S=\angle B C S=60^{\circ}$, so triangle $B C S$ is equilateral, and thus

(4) $B C=B S$.

Since $2 B S=B D$, it follows that $B D=2 B C$. From (1) and (4), we get $A C=B D=2 B C$.