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Circumcevian Triangle

Given a triangle and a point not a vertex of , define the -vertex of the circumcevian triangle as the point other than in which the line meets the circumcircle of , and similarly for and . Then is called the circumcevian triangle of (Kimberling 1998, p. 201).

The circumcevian triangle with respect to the point has trilinear vertex matrix

 (1)

and area

 (2)

where is the area of and

 (3)

Circumcevian triangles for various choices of are summarized in the table below.

 Kimberling circumcevian point circumcevian triangle incenter circumcircle mid-arc triangle triangle centroid circum-medial triangle orthocenter circum-orthic triangle

Every triangle inscribed in the circumcircle of a reference triangle is congruent to exactly one circumcevian triangle of (Kimberling 2005).

The circumcevian triangle of is similar to the pedal triangle of (Kimberling 1998), and it is homothetic to iff lies on the M'Cay cubic. The homothetic center lies on the Lemoine cubic (Gibert).

Circum-Medial Triangle, Circum-Orthic Triangle

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References

Gibert, B. "Lemoine Cubic." http://perso.wanadoo.fr/bernard.gibert/Exemples/k009.html.Gibert, B. "McCay Cubic = Griffiths Cubic." http://perso.wanadoo.fr/bernard.gibert/Exemples/k003.html.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Transfigured Triangle Geometry." Preprint. Mar. 5, 2005.

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Circumcevian Triangle

Cite this as:

Weisstein, Eric W. "Circumcevian Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CircumcevianTriangle.html