Circum-Orthic Triangle


The circum-orthic triangle DeltaA^'B^'C^' is the circumcevian triangle of a triangle DeltaABC with respect to the orthocenter H (Kimberling 1998, p. 163). The circum-orthic triangle of a triangle DeltaABC is also the image of the orthic triangle in the homothecy centered at the orthocenter H of DeltaABC and having similitude ratio 2.

It has trilinear vertex matrix

 [-ayz (by+cz)z (by+cz)y; (cz+ax)z -bzx (cz+ax)x; (ax+by)y (ax+by)x -cxy],

where x=cosA, y=cosB, and z=cosC.

Its area is


where Delta is the area of DeltaABC.

The following table gives the centers of the circum-orthic triangle in terms of the centers of the reference triangle corresponding to Kimberling centers X_n.

See also

Circumcevian Triangle, Medial Triangle

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Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Circum-Orthic Triangle

Cite this as:

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

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