Orthocenter
The intersection 
of the three Altitudes of a Triangleis called the orthocenter. Its Trilinear Coordinates are
 | (1) |
to give | (2) |
The Circumcenter 
and orthocenter are Isogonal Conjugate points. The orthocenter lies on the Euler Line.
 | (3) |
 | (4) |
 | (5) |
is the Inradius and 
is the Circumradius (Johnson 1929, p. 191).Any Hyperbola circumscribed on a Triangle and passing through the orthocenter isRectangular, and has its center on the Nine-Point Circle (Falisse 1920, Vandeghen1965).
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.
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