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GeezerVison is showing mathworld.wolfram.com — Outer Soddy Circle -- from Wolfram MathWorld

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Outer Soddy Circle

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OuterSoddyCircle

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

l=-((s-a)(s-a+2R_(S^')))/(bc),
(1)

where a, b, and c are the side lengths, s is the semiperimeter, and R_(S^') is the signed radius given below.

The signed radius of the outer Soddy circle is

R_(S^')=Delta/(4R+r-2s)
(2)
=(rs)/(4R+r-2s)
(3)
=(4r^2s)/([2(ab+bc+ca)-(a^2+b^2+c^2)]-8rs)
(4)
=(4Deltar)/([2(ab+bc+ca)-(a^2+b^2+c^2)]-8Delta)
(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; Jackson 2013). Here omega is the Brocard angle.

The geometric radius is |R_(S^')|. 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 4R+r=2s, 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 X_(175) (Kimberling 1994), which has equivalent 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 and S_A=bccosA is Conway triangle notation.

No notable triangle centers lie on the outer Soddy circle.

See also

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

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References

Dergiades, N. "The Soddy Circles." Forum Geometricorum 7, 191-197, 2007. https://web.archive.org/web/20231122034728/https://forumgeom.fau.edu/FG2007volume7/FG200726index.html.Jackson, F. M. "Soddyian Triangles." Forum Geometricorum 13, 1-6, 2013. https://www.researchgate.net/publication/265504822_Soddyian_triangles.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 Resource. https://mathworld.wolfram.com/OuterSoddyCircle.html

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