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Soddy Centers


SoddyCircles

Given three mutually tangent circles, there exist exactly two nonintersecting circles which are tangent circles to all three original circles. These are called the inner and outer Soddy circles, and their centers S and S^' are called the inner and outer Soddy centers, respectively.

The 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)
(1)
alpha_(176)=1+(bc)/(2(-a+b+c)R)
(2)
alpha_(176)=1+(r_A)/a,
(3)

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

The outer Soddy center S^' is the isoperimetric point X_(175), which has equivalent triangle center functions

alpha_(175)=-1+sec(1/2A)cos(1/2B)cos(1/2C)
(4)
alpha_(175)=-1+(bc)/(2(-a+b+c)R)
(5)
alpha_(175)=-1+(r_A)/a.
(6)

See also

Equal Detour Point, Inner Soddy Center, Inner Soddy Circle, Outer Soddy Center, Outer Soddy Circle, Isoperimetric Point, Soddy Circles

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References

Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, p. 181, 1994.

Referenced on Wolfram|Alpha

Soddy Centers

Cite this as:

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

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