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)
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where and are 8th-order and 16th-order polynomials, respectively.
The radius of the inner Soddy circle is
(2)
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(3)
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(4)
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(5)
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(6)
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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)
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(9)
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where is the circumradius of the reference triangle and is the -exradius.
It has circle function
(10)
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(P. Moses, pers. comm., Feb. 25, 2005), where , , and are the exradii.
No notable triangle centers lie on the inner Soddy circle.