285. A circle of constant radius moves, touching the edges of a trihedral angle, all plane angles of which are right angles. Find the geometric locus of the centers of these circles.
285. Introduce a rectangular coordinate system, taking the origin $O$ at the vertex of the trihedral angle, and directing the axes along the edges of this angle. Let the plane of the circle form angles $\alpha, \beta$, and $\gamma$ with the coordinate planes $X O Y, Y O Z$, and $Z O X$. Then the point $O_{1}$ - the center of the circle - will have coordinates $(R \sin \beta, R \sin \gamma, R \sin \alpha)$, where $R$ is the radius of the circle. Draw a line from the origin perpendicular to the plane of the circle. This line will form angles $\beta, \gamma$, and $\alpha$ with the axes. Therefore,

$$
\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1
$$

and, hence,

$$
\left|O O_{1}\right|^{2}=R^{2}\left(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma\right)=2 R^{2}
$$

Thus, the point $O_{1}$ lies on the surface of a sphere with center at $O$ and radius $R \sqrt{2}$. On the other hand, the distance from $O_{1}$ to the coordinate planes does not exceed $R$.

Therefore, the desired set represents a spherical triangle bounded by the planes $x=R, y=R$, $z=R$ on the surface of the sphere $\left|O O_{1}\right|=R \sqrt{2}$, located in the first octant.