Soddy Circles

Given three noncollinear points, construct three tangent circles such that one is centered at each point and the circles are pairwise tangent to one another. Then there exist exactly two nonintersecting circles that are tangent to all three circles. These are called the inner and outer Soddy circles, and their centers are called the inner and outer Soddy centers , respectively.

Frederick Soddy (1936) gave the formula for finding the radii of the Soddy circles () given the radii (, 2, 3) of the other three. The relationship is

(1)

where are the so-called bends, defined as the signed curvatures of the circles. If the contacts are all external, the signs are all taken as positive, whereas if one circle surrounds the other three, the sign of this circle is taken as negative (Coxeter 1969). Using the quadratic formula to solve for , expressing in terms of radii instead of curvatures, and simplifying gives

(2)

Here, the negative solution corresponds to the outer Soddy circle and the positive one to the inner Soddy circle.

The three lines through opposite points of tangency of any four mutually tangent circles are coincident, where "opposite" means here that the two circles determining one point of tangency are distinct from the two circles determining the other (Eppstein 2001). This fact gives rise to the first and second Eppstein points.

This formula is called the Descartes circle theorem since it was known to Descartes. Soddy extended the result to tangent spheres, and Gosper has further extended the result to mutually tangent -dimensional hyperspheres.

Bellew has derived a generalization applicable to a circle surrounded by circles which are, in turn, circumscribed by another circle. The relationship is

(3)

where is the curvature of the central circle,

(4)

and

(5)

For , this simplifies to the Descartes circle theorem

(6)