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Inner Soddy Circle


InnerSoddyCircle

The inner Soddy circle is the circle tangent to each of the three mutually tangent circles centered at the vertices of a reference triangle. 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 inner Soddy circle is

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

where Delta is the area of the reference triangle, 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 inner Soddy center, is the equal detour point X_(176) (Kimberling 1994), which has identical triangle center functions

alpha_(176)=1+sec(1/2A)cos(1/2B)cos(1/2C)
(7)
=1+(bc)/(2(-a+b+c)R)
(8)
=1+(r_A)/a,
(9)

where R is the circumradius of the reference triangle and r_A is the A-exradius.

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)]
(10)

(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 inner Soddy circle.


See also

Four Coins Problem, Inner Soddy Center, Outer Soddy Circle, 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.

Referenced on Wolfram|Alpha

Inner Soddy Circle

Cite this as:

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

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