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:
.