Soddy's hexlet

Soddy's hexlet is the mirror image.

Two spheres which are green and red are connected.

There are 6 spheres which are also connected each other around the two of them.

OP*OP'=R^2

OP and OP' are spheres, and R is the radius of OP'.
Therefore, OP=OP'=R
This is absurd, but P' is point at infinity.

The same spheres are connected, and they are mirror image.
Moreover, 6 spheres around OP and OP' are also mirror image.

∴

1/r1+1/r4=1/r2+1/r5=1/r3+1/r6