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Outer Soddy Circle


OuterSoddyCircle

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),
(1)

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

The radius of the outer Soddy circle is

R_(S^')=Delta/(4R+r+2s)
(2)
=(rs)/(4R+r+2s)
(3)
=(4r^2s)/(8rs-[2(ab+bc+ca)-(a^2+b^2+c^2)])
(4)
=(4Deltar)/(8Delta-[2(ab+bc+ca)-(a^2+b^2+c^2)])
(5)
=(S^2)/(2s[2s^2-(a^2+b^2+c^2)-2S])
(6)
=(S^2)/(4s[s^2-S(1+cotomega)]),
(7)

where Delta is the area of the reference triangle, R is the circumradius, 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 outer Soddy center, is the isoperimetric point X_(175) (Kimberling 1994), which has identical triangle center functions

alpha_(175)=-1+sec(1/2A)cos(1/2B)cos(1/2C)
(8)
=-1+(bc)/(2(-a+b+c)R)
(9)
=-1+(r_A)/a,
(10)

where R is the circumradius and r_A is the A-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)]
(11)

(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 outer Soddy circle.


See also

Inner Soddy Circle, Outer Soddy Center, Soddy Circles, Tangent Circles

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References

Dergiades, N. "The Soddy Circles." Forum Geometricorum 7, 191-197, 2007. http://forumgeom.fau.edu/FG2007volume7/FG200726index.html.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

Referenced on Wolfram|Alpha

Outer Soddy Circle

Cite this as:

Weisstein, Eric W. "Outer Soddy Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OuterSoddyCircle.html

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