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First Neuberg Circle

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FirstNeubergCircle

The first Neuberg circle is the circumcircle of the first Neuberg triangle. The center has center function

alpha=(-c^2b^6+a^4b^4+a^2c^2b^4-c^6b^2+a^2c^4b^2+a^4c^2b^2+a^4c^4)/a,
(1)

which is not a Kimberling center. Its radius is

R_N=(sqrt(f(a,b,c)f(b,c,a)f(c,a,b)))/(16Delta^2(a^2b^2+c^2b^2+a^2c^2)),
(2)

where

f(a,b,c)=a^6+b^2a^4+b^4a^2-b^2c^2a^2+b^6
(3)

and Delta is the area of the reference triangle.

The circle function is

l=-(a^2b^8-c^4b^6+a^6b^4-c^6b^4-a^2c^4b^4-a^4c^2b^4-a^4c^4b^2+a^6c^2b^2+a^2c^8+a^6c^4)/(b(a-b-c)(a+b-c)c(a-b+c)(a+b+c)(a^2b^2+c^2b^2+a^2c^2)),
(4)

which does not correspond to a Kimberling center.

No Kimberling centers lie on the first Neuberg circle.

See also

First Neuberg Triangle, Second Neuberg Circle

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Cite this as:

Weisstein, Eric W. "First Neuberg Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstNeubergCircle.html

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