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.

Given three mutually externally tangent circles, the inner Soddy circle lies in their curvilinear triangular gap and is tangent to all three. The Descartes' circle theorem determines its radius from the three given radii.