The Descartes circle theorem, also called the kissing circles problem, is a special case of Apollonius' problem requiring the determination of a circle touching three mutually tangent circles. There are two solutions: a small circle surrounded by the three original circles, and a large circle surrounding the original three.
Frederick Soddy gave the formula for finding the radius of the so-called inner and outer Soddy circles given the radii of the other three. The relationship is
 |
(1)
|
where 
are the curvatures of the circles
with radii 
.
Here, the negative solution corresponds to the outer
Soddy circle and the positive
solution to the inner Soddy circle.
This formula was known to Descartes and Viète (Boyer and Merzbach 1991, p. 159), but Soddy extended it to spheres. In 
-dimensional space, 
mutually touching 
-spheres can always be found, and
the relationship of their curvatures is
 |
(2)
|
A generalization of the theorem to the case of an "
-flower" consisting of 
tangent circles around the exterior of a central circle was
given by Mathews and Zymaris (2025), who also derived an explicit equation satfied
by their curvatures. In particular, let 
be the curvature of
a central circle and 
be the curvatures for 
, ..., 
surrounding mutually tangent circles forming an 
-flower. Define
 |  |  |
(3)
|
 |  |  |
(4)
|
for 
.
Then for 
odd,
![(m_0^2i)/2[product_(j=1)^(n-1)(m_j-i)-product_(j-1)^(n-1)(m_j+1)]-product_(j=1)^((n-1)/2)(m_(2j-1)^2+1)=0](/images/equations/DescartesCircleTheorem/NumberedEquation3.svg) |
(5)
|
and for 
even
![i/2[product_(j-1)^(n-1)(m_j-i)-product_(j=1)^(n-1)(m_j+i)]-product_(i=1)^(n/2-1)(m_(2j)^2+1)=0,](/images/equations/DescartesCircleTheorem/NumberedEquation4.svg) |
(6)
|
where 
and 
(Mathews and Zymaris 2025).
