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!
See also
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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.
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Cite this as:
Weisstein, Eric W. "Thomson Cubic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ThomsonCubic.html