Let be the orthic
triangle of a triangle . Then each side of each triangle meets the three sides
of the other triangle, and the points of intersection lie on a line called the orthic axis of .
The orthic axis is central line , has trilinear equation
It is perpendicular to the Euler
It passes through Kimberling centers for , 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,
See alsoAntiorthic Axis
, Orthic Triangle
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ReferencesCasey, 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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Cite this as:
Weisstein, Eric W. "Orthic Axis." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrthicAxis.html