Thomson cubic

See the following links for a detailed description:

Steve Phelps' visualization via barycentric coordinates
Bernard Gibert's web page on the Thomson cubic (see property 27)

It is notable that several well-known points of a triangle lie on its Thomson cubic, including the vertices, the midpoints of the sides, the circumcenter, the incenter, the excenters, the centroid and the orthocenter, to mention just the most important ones. Another quick way to create the Thomson cubic is to use GeoGebra's ImplicitCurve command. Usually, for a cubic curve one needs to give 9 different points of the curve to unequivocally define it. (This is true in general only under some non-degeneracy conditions, see Cramer's paradox.) So, the following command will immediately create the Thomson cubic: ImplicitCurve({A, B, C, (A+B)/2, (B+C)/2, (C+A)/2, (A+B+C)/3, Center(Circle(A, B, C)), Center(Incircle(A, B, C))}). (Don't forget to create the triangle ABC first! Be careful: isosceles triangles are considered a degenerate case.)

Acknowledgment

Bernard Gibert kindly pointed me to Cramer's paradox to have a better understanding what is going on in the degenerate case.

Thomson cubic

Acknowledgment

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