Orthic Axis


Let DeltaH_AH_BH_C be the orthic triangle of a triangle DeltaABC. Then each side of each triangle meets the three sides of the other triangle, and the points of intersection lie on a line O_AO_BO_C called the orthic axis of DeltaABC.

The orthic axis is central line L_3, has trilinear equation


It is perpendicular to the Euler line.

It passes through Kimberling centers X_i for i=230, 232, 468, 523 (isogonal conjugate of the focus of the Kiepert hyperbola), 647, 650, 676, 1637, 1886, 1990, 2485, 2489, 2490, 2491, 2492, 2493, 2501, 2506, 2977, 3003, 3011, 3012, and 3018. The anticomplement of the orthic axis is the de Longchamps line.

The orthic axis is the perspectrix of the medial triangle and tangential triangle, as well as (by definition) the orthic triangle and reference triangle.


It is the radical line of the coaxal system consisting of (circumcircle, nine-point circle, orthocentroidal circle, orthoptic circle of the Steiner inellipse, polar circle, tangential circle). This includes the particular cases of the circumcircle and the nine-point circle (Casey 1888, p. 176; Kimberling 1998, p. 150), as well as of any two of the circumcircle, nine-point circle, and polar circle (Tummers 1960-61).

The angle between the orthic axis and Gergonne line is equal to that between the Euler line and the Soddy line (F. Jackson, pers. comm., Nov. 2, 2005).

See also

Antiorthic Axis, Coaxal System, Orthic Triangle

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Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Honsberger, R. §13.2 (ii) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 151, 1995.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Tummers, J. H. "Zes merkwaardige punten die óók tot de negenpuntscirkel behoren." Nieuw Tijdschr. Wisk. 49, 250-252, 1960-61.

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Orthic Axis

Cite this as:

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

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