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The intersection $H$ of the three Altitudes of a Triangle is called the orthocenter. Its Trilinear Coordinates are

\cos B\cos C:\cos C\cos A:\cos A\cos B.
\end{displaymath} (1)

If the Triangle is not a Right Triangle, then (1) can be divided through by $\cos A\cos B\cos C$ to give
\sec A:\sec B:\sec C.
\end{displaymath} (2)

If the triangle is Acute, the orthocenter is in the interior of the triangle. In a Right Triangle, the orthocenter is the Vertex of the Right Angle.

\begin{figure}\begin{center}\BoxedEPSF{CircumcenterOrthocenter.epsf scaled 800}\end{center}\end{figure}

The Circumcenter $O$ and orthocenter $H$ are Isogonal Conjugate points. The orthocenter lies on the Euler Line.

\end{displaymath} (3)

\end{displaymath} (4)

\end{displaymath} (5)

where $r$ is the Inradius and $R$ is the Circumradius (Johnson 1929, p. 191).

Any Hyperbola circumscribed on a Triangle and passing through the orthocenter is Rectangular, and has its center on the Nine-Point Circle (Falisse 1920, Vandeghen 1965).

See also Centroid (Triangle), Circumcenter, Euler Line, Incenter, Orthic Triangle, Orthocentric Coordinates, Orthocentric Quadrilateral, Orthocentric System, Polar Circle


Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed. New York: Barnes and Noble, pp. 165-172, 1952.

Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 622, 1970.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 36-40, 1967.

Dixon, R. Mathographics. New York: Dover, p. 57, 1991.

Falisse, V. Cours de géométrie analytique plane. Brussels, Belgium: Office de Publicité, 1920.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 165-172 and 191, 1929.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Kimberling, C. ``Orthocenter.''

Vandeghen, A. ``Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.'' Amer. Math. Monthly 72, 1091-1094, 1965.

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© 1996-9 Eric W. Weisstein