Thomson Cubic


The Thomson cubic Z(X_2) of a triangle DeltaABC 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 x=bc 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 A, B, C, the side midpoints M_A, M_B, M_C, the altitude midpoints M_(H_A), M_(H_B), M_(H_C), the excenters J_A, J_B, J_C, the incenter I (X_1), triangle centroid G (X_2), circumcenter O (X_3), orthocenter H (X_4), and symmedian point K (X_6). It also passes through the mittenpunkt (X_9), as well as Kimberling centers X_(57), X_(223), and X_(282) (Kimberling 1998, p. 240), as well as X_(1073) and X_(1249) so it is really a 23-point cubic!

See also

Triangle Cubic

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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.", 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.

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Thomson Cubic

Cite this as:

Weisstein, Eric W. "Thomson Cubic." From MathWorld--A Wolfram Web Resource.

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