10,11

A sphere is placed in a trihedral angle, all plane angles of which are equal to $\alpha$, such that it touches all the edges of the trihedral angle. The faces of the trihedral angle intersect the sphere in circles of radius $r$. Find the radius of the sphere.
Let a sphere with center $O$ and radius $R$ touch the edges of a trihedral angle with vertex $P$ at points $A, B$, and $C$, and a circle with radius $r$ touches the lines $A P$ and $B P$ at points $A$ and $B$ respectively. Then

$$
a=A B=2 r \cos \frac{\alpha}{2}
$$

Consider a regular triangular pyramid $P A B C$ with vertex $P$. Let $P M$ be its height, $A B=B C=A C=a$. Denote by $\phi$ the angle between the height $P M$ and a lateral edge. Then

$$
\begin{aligned}
& A P=\frac{\frac{1}{A} a}{\operatorname{dn} \frac{\pi}{2}}, A M=\frac{1}{3} a \sqrt{3}, \sin \phi=\frac{A M}{A P}=\frac{2}{\sqrt{3}} \sin \frac{\alpha}{2}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_b09eb9152e3973547418g-40.jpg?height=120&width=684&top_left_y=1295&top_left_x=687)

Let $O 1$ be the center of the circle inscribed in the angle $A P B$. Then $O 1 A \perp A P$, so by the theorem of three perpendiculars, $O A \perp A P$. From the right triangle $A O P$, we find that

![](https://cdn.mathpix.com/cropped/2024_05_06_b09eb9152e3973547418g-40.jpg?height=123&width=1036&top_left_y=1522&top_left_x=511)

## Answer