12.010. An isosceles trapezoid with an acute angle \(\alpha\) at the base is circumscribed around a circle of radius \(R\). Find the perimeter of this trapezoid.
## Solution.

Given that $A B C D$ is a trapezoid, $A B=C D, O$ is the center of the circle inscribed in the trapezoid, $O E \perp A D, O E=R, \angle B A D=\alpha, \alpha<\pi / 2$ (Fig. 12.13); we need to find $P_{A B C D}=A D+2 A B+B C$. By the property of a circumscribed quadrilateral, $A D+B C=2 A B$ and $P_{A B C D}=4 A B$. Draw $B K \| O E$; then $B K=2 O E=2 R$. From $\triangle B K A$ we find $A B=\frac{2 R}{\sin \alpha}$ and $P_{A B C D}=\frac{8 R}{\sin \alpha}$.

Answer: $\frac{8 R}{\sin \alpha}$.