Sphere packing in a sphere

Question

I don't know how to prove that maximum radius of inner spheres equals $2\sqrt{3}-3$. I can't imagine it. Could you help me by any tips?

Result: https://en.wikipedia.org/wiki/Sphere_packing_in_a_sphere

Answer 1

Actually, it's the same, though parameters, indicated in the tables, are different: Maximum radius of inner spheres r, 1-st Table, Enclosing circle diameter R, 2-nd Table.

r=(2√3-3)R

If r=1, then

R=r/(2√3-3)=1/(2√3-3)=(2√3+3)/(12-9)=1+2/√3

Sorry for non-decent format, I'm a rookie in that.

Answer 2

Piece of cake. Let's set the plane consisting 3 centers of inner circles. This plane consists the center of large circle (easy proof by central axis symmetry). Then, you have equilateral triangle, each side of which is $2r$, its center coincides with center of large sphere, $R=1$.

Hence, you have an equation for the segment between the triangle's vertex and its center: $$1-r=\frac{2r}{\sqrt{3}}$$
$$r=\frac{\sqrt{3}}{2+\sqrt{3}}=2\sqrt{3}-3$$