Thomson Cubic

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A circumconic is a conic section through the vertices of a triangle [1].

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Given a triangle , the Thomson cubic of is the set of the centers of circumconics whose normals at the vertices are concurrent [2]. It is a self-isogonal cubic with pivot point at the triangle centroid.

Let , , be the side lengths of the reference triangle and , , be the excenters. Then the equation of the Thomson cubic of triangle in barycentric coordinates is

, where indicates that the sum is taken over all six permutations of , , .

The solution is traced in red. Some of the Kimberling centers on the Thomson cubic are:

, , , , , , , , , , , , , , , , , [3].

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, "Triangle Centers and Central Triangles." Congressus Numerantium, 129, 1–295, 1998.

[2] B. Gilbert. "Thomson Cubic = pK(X6,X2)." (Jul 20, 2022) bernard-gibert.pagesperso-orange.fr/Exemples/k002.html.

[3] Encyclopedia of Triangle Centers (ETC). https://faculty.evansville.edu/ck6/encyclopedia/etc.html.

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

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