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