## Task B-3.4.

Calculate the area of rhombus $ABCD$ if the radii of the circumscribed circles of triangles $ABD$ and $ACD$ are 10 and 20, respectively.
## Solution.

Let the radius of the circumcircle of triangle $A B D$ be denoted by $r$, and the radius of the circumcircle of triangle $A C D$ by $R$.

By the Law of Sines for triangles $A B D$ and $A C D$, we have:

$$
\frac{e}{\sin 2 \alpha}=2 r=20, \quad \frac{f}{\sin \left(180^{\circ}-2 \alpha\right)}=2 R=40
$$

Thus, $e=20 \sin 2 \alpha, f=40 \sin 2 \alpha$.

![](https://cdn.mathpix.com/cropped/2024_05_30_6dc7744662d888032875g-20.jpg?height=642&width=1245&top_left_y=250&top_left_x=334)

Then,

$$
\frac{e}{f}=\frac{20 \sin 2 \alpha}{40 \sin 2 \alpha}=\frac{1}{2}
$$

Thus, $f=2 e$,

and $a=|A D|=\sqrt{\left(\frac{e}{2}\right)^{2}+\left(\frac{f}{2}\right)^{2}}=\frac{e \sqrt{5}}{2}$,

from which it follows that $\sin \alpha=\frac{1}{\sqrt{5}}, \cos \alpha=\frac{2}{\sqrt{5}}$

and $\sin 2 \alpha=\frac{4}{5}$

1 point

It follows that $e=20 \sin 2 \alpha=20 \cdot \frac{4}{5}=16, f=32$.

1 point

The area of the rhombus is $P=\frac{e \cdot f}{2}=\frac{16 \cdot 32}{2}=256$.

1 point