Four touching Apollonius circles from inverse square law ?

From: Narasimham G.L. (mathma18_at_hotmail.com)
Date: 12/25/04 

Date: 25 Dec 2004 09:33:07 -0800

Let us assume inverse squared law of gravity force F = G m M /R^2 = C
k^2 is operative between two bodies of mass m,M reperesented pairwise
by circles of radius (R1,R2,R3,R4) or their reciprocal curvatures
(k1,k2,k3,k4). The circle of interaction extends upto their radius
limit at respective points of tangency (R1,R2,R3,R4). One can be less
vague here, but...

Is it somehow possible to show/derive by force equilibrium :
(k1+k2+k3+k4)^2 = 2*(k1^2+k2^2+k3^2+k4^2)^2 ?