10.070. A circle is inscribed in a circular sector with a central angle of $120^{\circ}$. Find the radius of the inscribed circle if the radius of the given circle is $R$.
Solution.

$\Delta O_{2} A O_{1}$ is a right triangle, $\angle O_{1} O_{2} A=60^{\circ}, \sin 60^{\circ}=\frac{O_{1} A}{O_{2} O_{1}}=\frac{r}{R-r} ;$ $\frac{\sqrt{3}}{2}=\frac{r}{R-r}$ (Fig. 10.69). From this, $r=\frac{\sqrt{3} R}{2+\sqrt{3}}=\sqrt{3} R(2-\sqrt{3})$.

Answer: $\sqrt{3} R(2-\sqrt{3})$.