Prove: I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then
$$ddeeff+ddeexx+ddffxx+eeffxx = \\2(deffxx+ddeffx+deefxx+ddeefx+ddefxx+deeffx)$$
[Reference: p.189-190 of The Changing Shape of Geometry.]
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