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What ratio should the radii of three spheres, each touching the other two, satisfy so that a common tangent plane can be drawn to all three?

#
Let $r_1 \geqslant r_2$ be the radii of spheres with centers $O_1$ and $O_2$ respectively. Consider an arbitrary plane tangent to both spheres at points $A$ and $B$ respectively. Through the parallel lines $O_1A$ and $O_2B$, draw a plane. In the section of this plane, we obtain touching circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ and radii $O_1A = r_1$ and $O_2B = r_2$, touching the line $AB$ at points $A$ and $B$. The radius $r_3$ of the smaller sphere touching the first two must be no less than the radius $r$ of the circle touching circles $S_1$ and $S_2$ and the segment $AB$ at some point $C$. Then $AC + BC = AB$, or

$$
2 \sqrt{r r_1} + 2 \sqrt{r r_2} = 2 \sqrt{r_1 r_2}
$$

from which $r = \frac{r_1 r_2}{\left(\sqrt{r_1} + \sqrt{r_2}\right)^2}$. It is clear that for any sphere of radius $r_3 \geqslant r$ touching the first two spheres, there exists a common tangent plane to all three spheres.

## Answer

If $r_3 \leqslant r_2 \leqslant r_1$, then $r_3 \geqslant \frac{r_1 r_2}{\left(\sqrt{r_1} + \sqrt{r_2}\right)^2}$.