Descartes' Circle Formula

From AoPSWiki

(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, $\frac{1}{r_a}$ and $\frac{1}{r_b}$ .

If circle $A$ is internally tangent to circle $B$ , however, a the curvature of circle $A$ is still $\frac{1}{r_a}$ , while the curvature of circle B is $-\frac{1}{r_b}$ , 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 the following equation holds:

$(a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2)$ .