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Third Lemoine Circle

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ThirdLemoineCircle

The third Lemoine circle, a term coined here for the first time, is the circumcircle of the Lemoine triangle.

It has center function

alpha=(f(a,b,c))/a,
(1)

where f(a,b,c) is a 10th-order polynomial, which is not a Kimberling center and radius

R=(sqrt((g(a,b,c)g(b,c,a)g(c,a,b))/((-a+b+c)(a+b-c)(a-b+c)(a+b+c))))/(2(4a^2+b^2+c^2)(a^2+4b^2+c^2)(a^2+b^2+4c^2)),
(2)

where

g(a,b,c)=-3a^6+b^2a^4+3c^2a^4+5b^4a^2+3c^4a^2+23b^2c^2a^2+b^6-3c^6+b^2c^4+5b^4c^2.
(3)

Its circle function is

l=-((a^2-2b^2-2c^2)(a^6+a^4b^2-a^2b^4-b^6+a^4c^2-17a^2b^2c^2-9b^4c^2-a^2c^4-9b^2c^4-c^6))/(2bc(4a^2+b^2+c^2)(a^2+4b^2+c^2)(a^2+b^2+4c^2)),
(4)

which is also not a Kimberling center.

It passes through X_(115), the center of the Kiepert hyperbola.

See also

Cosine Circle, First Lemoine Circle, Lemoine Triangle

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Cite this as:

Weisstein, Eric W. "Third Lemoine Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ThirdLemoineCircle.html

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