Third Lemoine Circle

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ThirdLemoineCircle

The name third Lemoine circle is used in this work for the circumcircle of the Lemoine triangle.

It has center function

(1)

where is a 10th-order polynomial, which is not a Kimberling center and radius

(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 , the center of the Kiepert hyperbola.

A generally different circle is called Ehrmann's third Lemoine circle (Grinberg 2012, pp. 41-42). The circles through the symmedian point and each pair of vertices of the reference triangle meet the lines containing the other two sides again in six concyclic points. This circle has center and radius , where is the circumcenter, is the circumradius, and is the radius of the cosine circle.

See also

Cosine Circle, First Lemoine Circle, Lemoine Triangle

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References

Grinberg, D. "Ehrmann's Third Lemoine Circle." J. Classical Geom. 1, 40-52, 2012. https://darijgrinberg.gitlab.io/JCG2012V1pp40-52.pdf.Kimberling, C. "Encyclopedia of Triangle Centers: X(576)=Orthoharmonic of X(187)." https://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X576.

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Third Lemoine Circle

Cite this as:

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

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