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.
![[asy] size(200); defaultpen(linewidth(0.7)); draw(Circle(origin,0.5)); draw(Circle((1.5,0),1)); dot(origin^^(1.5,0)^^(0.5,0)); draw(origin--(1.5,0)); label("$1/2$", (0.25,0), N); label("$1$", (1,0), N); label("$A$", origin, SW); label("$B$", (1.5,0), SE); [/asy]](http://latex.artofproblemsolving.com/a/1/0/a10298bb91d00b0aaace608317f8d041d7040165.png)
In the above diagram, the curvature of circle A is 2 while the curvature of circle B is 1.
![[asy] size(200); defaultpen(linewidth(0.7)); draw(Circle((1.25,0),0.25)); draw(Circle((1.5,0),0.5)); dot((1,0)^^(1.5,0)^^(1.25,0)^^(2,0)); draw((1,0)--(2,0)); label("$1/2$", (1.125,0), N); label("$1$", (1.75,0), N); label("$A$", (1.25,0), SW); label("$B$", (1.5,0), SE); [/asy]](http://latex.artofproblemsolving.com/a/f/4/af47ea9e3ec1c5e986ebbc66c01ff3e06addce01.png)
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:
.
