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# Thomson Cubic

The Thomson cubic of a triangle is the locus the centers of circumconics whose normals at the vertices are concurrent. It is a self-isogonal cubic with pivot point at the triangle centroid, so its parameter is and its trilinear equation is given by

(Cundy and Parry 1995; Kimberling 1998, p. 240).

It is sometimes called the seventeen-point cubic (Casey 1893, p. 460; Kimberling 1998, p. 240) because it passes through the vertices , , , the side midpoints , , , the altitude midpoints , , , the excenters , , , the incenter (), triangle centroid (), circumcenter (), orthocenter (), and symmedian point (). It also passes through the mittenpunkt (), as well as Kimberling centers , , and (Kimberling 1998, p. 240), as well as and so it is really a 23-point cubic!

Triangle Cubic

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## References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, and Co., p. 460, 1893.Cundy, H. M. and Parry, C. F. "Some Cubic Curves Associated with a Triangle." J. Geom. 53, 41-66, 1995.Gibert, B. "Thomson Cubic." http://perso.wanadoo.fr/bernard.gibert/Exemples/k002.html.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Rubio, P. "Cubic Lines Relative to a Triangle." J. Geom. 34, 152-171, 1989.Thomson, F. D. Educ. Times. Aug. 1864.

Thomson Cubic

## Cite this as:

Weisstein, Eric W. "Thomson Cubic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ThomsonCubic.html