The Soddy circles are the two nonintersecting circles tangent to each of three pairwise externally tangent
circles centered at noncollinear points. In the limiting
case when the three circles have a common tangent line, the outer solution is that
line, regarded as a circle of infinite radius (Eppstein
2001). They are called the inner Soddy circle
and outer Soddy circle. When both solutions
are circles, their centers are called the inner
Soddy center
and outer Soddy center, respectively.
Frederick Soddy (1936) gave the formula for finding the radii of the Soddy circles () given the radii (,
2, 3) of the other three. The relationship is
(1)
where
are the so-called bends, defined as the signed curvatures
of the circles. If the contacts are all external, the
signs are all taken as positive, whereas if one circle
surrounds the other three, the sign of this circle is taken as negative
(Coxeter 1969). Using the quadratic formula
to solve for ,
expressing in terms of radii instead of curvatures, and simplifying gives
(2)
Here, the minus branch corresponds to the outer Soddy circle and the plus branch to the inner
Soddy circle. The signed radius of the outer
Soddy circle is negative when it encloses the other three circles and positive
when all three tangencies are external. A zero denominator represents the limiting
tangent line.
The three lines through opposite points of tangency of any four mutually tangent circles are concurrent, where "opposite"
means here that the two circles determining one point of tangency are distinct from
the two circles determining the other (Eppstein 2001). This fact gives rise to the
first and second
Eppstein points.
This formula is called the Descartes circle theorem since it was known to Descartes. The higher-dimensional generalization
concerns
mutually tangent -dimensional
hyperspheres (Anonymous 1937).
Bellew has derived a generalization applicable to a circle surrounded by circles which are, in turn, circumscribed by another circle.
The relationship is
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and outer Soddy center 
, respectively.
) given the radii 
(
,
2, 3) of the other three. The relationship is
are the so-called bends, defined as the signed curvatures
of the circles. If the contacts are all external, the
signs are all taken as positive, whereas if one circle
surrounds the other three, the sign of this circle is taken as negative
(Coxeter 1969). Using the quadratic formula
to solve for 
,
expressing in terms of radii instead of curvatures, and simplifying gives
mutually tangent 
-dimensional
hyperspheres (Anonymous 1937).
circles which are, in turn, circumscribed by another circle.
The relationship is
![[n(c_n-1)^2+1]sum_(i=1)^(n+1)kappa_i^2+n(3nc_n^2-2n-6)c_n^2(c_n-1)^2=[(f(n))/(n(c_n-1)+1)]^2,](/images/equations/SoddyCircles/NumberedEquation3.svg)
is the curvature of the central circle,
![f(n)=[n(c_n-1)^2+1]sum_(i=1)^(n+1)kappa_i+nc_n(c_n-1)[nc_n^2+(3-n)c_n-4]](/images/equations/SoddyCircles/NumberedEquation4.svg)
,
this simplifies to the Descartes circle theorem
Dimensions."