The inner Soddy circle is the circle tangent to each of the three mutually tangent circles centered at the vertices of a reference triangle. It has circle function
(1)

where and are 8thorder and 16thorder polynomials, respectively.
The radius of the inner Soddy circle is
(2)
 
(3)
 
(4)
 
(5)
 
(6)

where is the area of the reference triangle, is its inradius, is the semiperimeter, and is Conway triangle notation (P. Moses, pers. comm., Feb. 25, 2005; Dergiades 2007).
Its center, known as inner Soddy center, is the equal detour point (Kimberling 1994), which has identical triangle center functions
(7)
 
(8)
 
(9)

where is the circumradius of the reference triangle and is the exradius.
It has circle function
(10)

(P. Moses, pers. comm., Feb. 25, 2005), where , , and are the exradii.
No notable triangle centers lie on the inner Soddy circle.