12.14. A circle of radius $R$ touches the edges of a given trihedral angle, all plane angles of which are right angles. Find the geometric locus of all possible positions of its center.

## § 3. Various Loci
12.14. Introduce a rectangular coordinate system, directing its axes along the edges of the given trihedral angle. Let $O_{i}$ be the center of the circle; П - the plane of the circle; $\alpha, \beta$ and $\gamma$ - the angles between the plane П and the coordinate planes. Since the distance from the point $O_{1}$ to the line of intersection of the planes П and $O y z$ is $R$, and the angle between these planes is $\alpha$, the distance from the point $O_{1}$ to the plane $O y z$ is $R \sin \alpha$. Analogous reasoning shows that the point $O_{1}$ has coordinates $(R \sin \alpha, R \sin \beta, R \sin \gamma)$. Since $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$ (problem 1.21), then $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$, and therefore, $O O_{1}=$ $=\sqrt{2} R$. Moreover, the distance from the point $O_{1}$ to any face of the trihedral angle is not greater than $R$. The desired locus of points is a part of a sphere of radius $\sqrt{2} R$ with the center at the origin, bounded by the planes $x=R, y=R$ and $z=R$.