Circumcevian Triangle


Given a triangle DeltaABC and a point P not a vertex of DeltaABC, define the A^'-vertex of the circumcevian triangle as the point other than A in which the line AP meets the circumcircle of DeltaABC, and similarly for B^' and C^'. Then DeltaA^'B^'C^' is called the circumcevian triangle of DeltaABC (Kimberling 1998, p. 201).

The circumcevian triangle DeltaA^'B^'C^' with respect to the point P=alpha:beta:gamma has trilinear vertex matrix

 [-abetagamma (bgamma+cbeta)beta (bgamma+cbeta)gamma; (calpha+agamma)alpha -bgammaalpha (calpha+agamma)gamma; (abeta+balpha)alpha (abeta+balpha)beta -calphabeta]

and area


where Delta is the area of DeltaABC and


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

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

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

See also

Circum-Medial Triangle, Circum-Orthic Triangle, Circumanticevian Triangle

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Gibert, B. "Lemoine Cubic.", B. "McCay Cubic = Griffiths Cubic.", C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Transfigured Triangle Geometry." Preprint. Mar. 5, 2005.

Referenced on Wolfram|Alpha

Circumcevian Triangle

Cite this as:

Weisstein, Eric W. "Circumcevian Triangle." From MathWorld--A Wolfram Web Resource.

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