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# 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 $r_a$ is externally tangent to circle B of radius $r_b$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_1}$ and $\frac{1}{r_2}$.

If circle A is internally tangent to circle B, however, a the curvature of circle A is still $\frac{1}{r_1}$, while the curvature of circle B is $-\frac{1}{r_2}$, the opposite of the reciprocal of its radius.

[asy] draw(Circle(origin,2)); [/asy]