Outer Soddy Circle

The outer Soddy circle is the second circle tangent to the three mutually tangent circles centered at the vertices of a reference triangle, in addition to the inner Soddy circle. Its circle function is

(1)

where , , and are the side lengths, is the semiperimeter, and is the signed radius given below.

The signed radius of the outer Soddy circle is

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

where is the area of the reference triangle, is the 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.

The geometric radius is . A negative signed radius means that the outer Soddy circle encloses the three vertex circles, while a positive one means that it is externally tangent to them. When , it degenerates into a straight line and the finite-radius formulas do not apply. The reference triangle is then a Soddyian triangle (Jackson 2013, pp. 1-2).

Its center, known as outer Soddy center, is the isoperimetric point (Kimberling 1994), which has equivalent triangle center functions

			(8)
			(9)
			(10)

where is the circumradius and is the -exradius of the reference triangle.

It has circle function

(11)

(P. Moses, pers. comm., Feb. 25, 2005), where , , and are the exradii and is Conway triangle notation.

No notable triangle centers lie on the outer Soddy circle.