Given three mutually tangent circles, there exist exactly two nonintersecting circles which are tangent circles
to all three original circles. These are called the inner
and outer Soddy circles, and their centers 
and 
are called the inner
and outer Soddy centers, respectively.
The inner Soddy center is the equal detour point 
(Kimberling 1994), which has identical triangle center functions
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the circumradius of the reference
triangle and 
is the 
-exradius.
The outer Soddy center 
is the isoperimetric
point 
,
which has equivalent triangle center functions
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
