## Orthocenter

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

 (1)

If the Triangle is not a Right Triangle, then (1) can be divided through by to give
 (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.

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

 (3)

 (4)

 (5)

where is the Inradius and 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

References

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.'' http://cedar.evansville.edu/~ck6/tcenters/class/orthocn.html.

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