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
![l=((-a+b+c)^2[f(a,b,c)-16g(a,b,c)rs])/(4bc[(a^2+b^2+c^2)-2(ab+bc+ca)-8rs]^4),](/images/equations/InnerSoddyCircle/NumberedEquation1.svg) |
(1)
|
where 
and 
are 8th-order and 16th-order polynomials, respectively.
The radius of the inner Soddy circle is
 |  |  |
(2)
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(3)
|
 |  | ![(4Deltar)/(8Delta+[2(ab+bc+ca)-(a^2+b^2+c^2)])](/images/equations/InnerSoddyCircle/Inline11.svg) |
(4)
|
 |  | ![(S^2)/(2s[2s^2-(a^2+b^2+c^2)+2S])](/images/equations/InnerSoddyCircle/Inline14.svg) |
(5)
|
 |  | ![(S^2)/(4s[s^2+S(1+cotomega)]),](/images/equations/InnerSoddyCircle/Inline17.svg) |
(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)
|
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(8)
|
 |  |  |
(9)
|
where 
is the circumradius of the reference
triangle and 
is the 
-exradius.
It has circle function
![l=1/(bc)[(S^4)/(16s^2[S(cotomega-1)-s^2]^2)-(c^2(b-r_B)^2+2(c-r_C)(b-r_B)S_A+b^2(c-r_C)^2)/((r_A+r_B+r_C-2s)^2)]](/images/equations/InnerSoddyCircle/NumberedEquation2.svg) |
(10)
|
(P. Moses, pers. comm., Feb. 25, 2005), where 
, 
, and 
are the exradii.
No notable triangle centers lie on the inner Soddy circle.
