= Soddy circle
{c}
Given three mutually externally tangent <circles> with disjoint interiors, a <Soddy circle> is a circle tangent to all three. The inner solution lies in the curvilinear triangular gap. In signed <curvatures> $b_i=\pm1/r_i$, the two possible fourth curvatures satisfy
$$
b_4=b_1+b_2+b_3\pm2\sqrt{b_1b_2+b_2b_3+b_3b_1}.
$$
A containing circle uses negative curvature. For three unit <circles>, the inner curvature is $3+2\sqrt3$, while the outer curvature $3-2\sqrt3<0$ describes a containing circle. The formula is the quadratic solution of the <Descartes' circle theorem>; zero curvature is interpreted as a tangent <line>.
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