Smallest tangent sphere to three mutually tangent spheres

The cross-section passing through the centers of the three spheres would be three mutually tangent circles, and the smallest sphere that is tangent to them would be another mutually tangent circle to the original three:

By Descartes' Theorem , the radius $r$ of the small sphere satisfies the equation:

$\bigg(\cfrac{1}{5} + \cfrac{1}{7} + \cfrac{1}{10} + \cfrac{1}{r}\bigg) = 2\bigg(\cfrac{1}{5^2} + \cfrac{1}{7^2} + \cfrac{1}{10^2} + \cfrac{1}{r^2}\bigg)$

which solves to $r = \cfrac{280\sqrt{77} - 2170}{271} \approx \boxed{1.059}$ .