 Soddy circle

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 $b_i=\pm 1/r_i$, the two possible fourth curvatures satisfyA containing circle uses negative curvature. For three unit circles, the inner curvature is $3+2\sqrt{3}$, while the outer curvature $3-2\sqrt{3}<0$ describes a containing circle. The formula is the quadratic solution of the Descartes' circle theorem; zero curvature is interpreted as a tangent line.