# Descartes' Circle Formula

(based on wording of ARML 2010 Power)

Descartes' Circle Formula is a relation held between four mutually tangent circles.

Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle A of radius is externally tangent to circle B of radius . Then the curvatures of the circles are simply the reciprocals of their radii, and .

If circle A is internally tangent to circle B, however, a the curvature of circle A is still , while the curvature of circle B is , the opposite of the reciprocal of its radius.

In the above diagram, the curvature of circle A is 2 while the curvature of circle B is 1.

In the above diagram, the curvature of circle A is still 2 while the curvature of circle B is -1.

When four circles A, B, C, and D are pairwise tangent, with respective curvatures a, b, c, and d, then:

.