Calculate the Soddy circles

1st circle :Radius a
2nd circle :Radius b
3rd circle :Radius c
(radius < 0 if circle surrounds the others)

 

Method

The radius d of the 4th circle is given by the Descartes formula  2(1/a² + 1/b² + 1/c² + 1/d²) = (1/a + 1/b + 1/c + 1/d)² 

which, solved in d, gives

 d = abc/(ab + bc + ac ± 2√abc(a + b + c)

This formula may give d infinite (the 4th circle is a straight line) or d < 0 (the 4th circle surrounds the others)
At most one of the 4 circles may surround the 3 other.
The script gives an exact value as (P +  u√v)/Q if a,b,c are integers
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