6. A circle with center $O_{1}$ and radius 2, inscribed in triangle $A B C$, touches side $B C$ at point $D$. A second circle with center $O_{2}$ and radius 4 touches the extensions of sides $A B$ and $A C$, as well as side $B C$ at point $E$. Find the area of quadrilateral $O_{1} D O_{2} E$, if the measure of angle $A C B$ is $120^{\circ} . \quad(20$ points)
# Solution.

$S=S_{\square O_{1} E D}+S_{\square O_{2} E D}=1 / 2 \cdot E D \cdot\left(O_{1} D+O_{2} E\right)=3 E D$

$E D=E C-D C=O_{2} E / \operatorname{tg} 30-O_{1} D / \operatorname{tg} 60=4 \sqrt{3}-2 / \sqrt{3}=10 / \sqrt{3}$ $S=30 / \sqrt{3}$

Answer: $30 / \sqrt{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c995faa73b19f5c426fdg-13.jpg?height=485&width=497&top_left_y=277&top_left_x=1519)

## Solution for Variant №10-9