Inner Soddy Circle

The inner Soddy circle is the circle externally 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 the side lengths, is the semiperimeter, and is the radius given below.

The radius of the inner Soddy circle is

			(2)
			(3)
			(4)
			(5)
			(6)

where is the area of the reference triangle, is its circumradius, is its inradius, is the semiperimeter, and is Conway triangle notation (P. Moses, pers. comm., Feb. 25, 2005; Dergiades 2007; Jackson 2013). Here is the Brocard angle.

Its center, known as inner Soddy center, is the equal detour point (Kimberling 1994), which has equivalent 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 and is Conway triangle notation.

No notable triangle centers lie on the inner Soddy circle.