The Kiss Precise
Four circles to the kissing come,
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of squares of all four bends
Is half the square of their sum.
Frederick Soddy, The Kiss Precise, the Hexlet, and the Bowl of Integers (1936)
Notes
If four circles A, B, C, and D, of radii r1, r2, r3, and r4, are drawn so that they do not overlap but each touches the other three, and if we let b1 = 1/r1, etc., then
(b1 b2 b3 b4)^2 = 2(b1^2 b2^2 b3^2 b4^2).
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