9. Given four balls of the same radius $r$, each touching the other three. Calculate the radius of the ball that touches all the given balls.
9. There will be two such spheres. One of them will touch the four spheres internally, and the other externally. Let the radius of the sphere touching internally be denoted by $p$, and the radius of the sphere touching externally by $R$. Then it is not difficult to show that the centers of the last two spheres coincide with the center of gravity of a regular tetrahedron, the vertices of which lie at the centers of the given four spheres. From this, we obtain

$$
\rho=r\left(\frac{\sqrt{6}}{2}-1\right) ; \quad R=r\left(\frac{\sqrt{6}}{2}+1\right)
$$

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