# 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 is internally tangent to circle , however, a the curvature of circle 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 is while the curvature of circle is .

In the above diagram, the curvature of circle is still while the curvature of circle is .

When four circles and are pairwise tangent, with respective curvatures and , then the following equation holds:

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