Inner Soddy Circle


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


where f(a,b,c) and g(a,b,c) are 8th-order and 16th-order polynomials, respectively.

The radius of the inner Soddy circle is


where Delta is the area of the reference triangle, r is its inradius, s is the semiperimeter, and S=2Delta 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 X_(176) (Kimberling 1994), which has identical triangle center functions


where R is the circumradius of the reference triangle and r_A is the A-exradius.

It has circle function


(P. Moses, pers. comm., Feb. 25, 2005), where r_A, r_B, and r_C are the exradii.

No notable triangle centers lie on the inner Soddy circle.

See also

Four Coins Problem, Inner Soddy Center, Outer Soddy Circle, Soddy Circles, Tangent Circles

Explore with Wolfram|Alpha


Dergiades, N. "The Soddy Circles." Forum Geometricorum 7, 191-197, 2007.

Referenced on Wolfram|Alpha

Inner Soddy Circle

Cite this as:

Weisstein, Eric W. "Inner Soddy Circle." From MathWorld--A Wolfram Web Resource.

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