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Neuberg Cubic

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Given a triangle , the Neuberg cubic is the set of all points whose reflections in the sides , and form a triangle perspective to . It is a self-isogonal cubic with pivot point at the Euler infinity point [1]. The name comes from the geometer Joseph Jean Baptiste Neuberg for his 1894 paper.

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Let , , be the side lengths of and let , , be the excenters of . Then the equation of the Neuberg cubic of in barycentric coordinates is given by

, where the cyclic sum is over all six permutations of , , .

The cubic passes through the points , , and the Kimberling centers , , , , , , , , , , , , .

You can drag the vertices , and .

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Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA

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References

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Aug 2, 2022) faculty.evansville.edu/ck6/encyclopedia.

[2] B. Gilbert. "Catalogue of Triangle Cubics." (Aug 3, 2022) https://bernard-gibert.pagesperso-orange.fr/ctc.html.

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