Orthocenter

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

$$\cos B\cos C:\cos C\cos A:\cos A\cos B.$$ (1)

If the Triangle is not a Right Triangle, then (1) can be divided through by $\cos A\cos B\cos C$ to give

$$\sec A:\sec B:\sec C.$$ (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 $O$ and orthocenter $H$ are Isogonal Conjugate points. The orthocenter lies on the Euler Line.

$${a_1}^2+{a_2}^2+{a_3}^2+\overline{A_1H}^2+\overline{A_2H}^2+\overline{A_3H}^2=12R^2$$ (3)

$$\overline{A_1H}+\overline{A_2H}+\overline{A_3H}=2(r+R),$$ (4)

$$\overline{A_1H}^2+\overline{A_2H}^2+\overline{A_3H}^2=4R^2-4Rr,$$ (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

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.