The outer Soddy circle is the solution to the four coins problem. 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/OuterSoddyCircle/NumberedEquation1.svg) |
(1)
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where 
and 
are 8th-order and 16th-order polynomials, respectively.
The radius of the outer Soddy circle is
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(2)
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(3)
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 |  | ![(4r^2s)/(8rs-[2(ab+bc+ca)-(a^2+b^2+c^2)])](/images/equations/OuterSoddyCircle/Inline11.svg) |
(4)
|
 |  | ![(4Deltar)/(8Delta-[2(ab+bc+ca)-(a^2+b^2+c^2)])](/images/equations/OuterSoddyCircle/Inline14.svg) |
(5)
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 |  | ![(S^2)/(2s[2s^2-(a^2+b^2+c^2)-2S])](/images/equations/OuterSoddyCircle/Inline17.svg) |
(6)
|
 |  | ![(S^2)/(4s[s^2-S(1+cotomega)]),](/images/equations/OuterSoddyCircle/Inline20.svg) |
(7)
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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).
Its center, known as outer Soddy center, is the isoperimetric point 
(Kimberling 1994), which has identical triangle center
functions
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(8)
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(9)
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(10)
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where 
is the circumradius and 
is the 
-exradius of the reference
triangle.
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/OuterSoddyCircle/NumberedEquation2.svg) |
(11)
|
(P. Moses, pers. comm., Feb. 25, 2005), where 
, 
, and 
are the exradii.
No notable triangle centers lie on the outer Soddy circle.
