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Orthocenter

The orthocenter is the intersection of the three altitudes , , and of a triangle. The name was invented by Besant and Ferrers in 1865 while walking on a road leading out of Cambridge, England in the direction of London (Satterly 1962). The trilinear coordinates of the orthocenter are

(1)

If the triangle is not a right triangle, then (1) can be divided through by to give

(2)

The orthocenter is Kimberling center .

The following table summarizes the orthocenters for named triangles that are Kimberling centers.

triangle	Kimberling	orthocenter
anticomplementary triangle		de Longchamps point
circumcircle mid-arc triangle		incenter
circumnormal triangle		circumcenter
circumtangential triangle		circumcenter
contact triangle		orthocenter of the contact triangle
Euler-Gergonne-Soddy triangle		Fletcher point
Euler triangle		orthocenter
excentral triangle		incenter
first Brocard triangle		reflection of in
first Morley triangle		first Morley center
first Yff circles triangle		center of the first Johnson-Yff circle
Fuhrmann triangle		incenter
hexyl triangle		Bevan point
incentral triangle		orthocenter of the incentral triangle
inner Napoleon triangle		triangle centroid
inner Vecten triangle		inner Vecten point
medial triangle		circumcenter
mid-arc triangle		first mid-arc point
orthic triangle		orthocenter of orthic triangle
outer Napoleon triangle		triangle centroid
outer Vecten triangle		outer Vecten point
reference triangle		orthocenter
second Yff circles triangle		(,)-harmonic conjugate of
Stammler triangle		circumcenter
Steiner triangle		circumcenter
tangential triangle		eigencenter of orthic triangle
Yff contact triangle		Nagel point

If the triangle is acute, the orthocenter is in the interior of the triangle. In a right triangle, the orthocenter is the vertex at the right angle.

When the vertices of a triangle are combined with its orthocenter, any one of the points is the orthocenter of the other three, as first noted by Carnot (Wells 1991). These four points therefore form an orthocentric system.

The circumcenter and orthocenter are isogonal conjugates.

The orthocenter lies on the Euler line. It lies on the Fuhrmann circle and orthocentroidal circle, and the orthocenter and Nagel point form a diameter of the Fuhrmann circle. It is the center of the polar circle and first Droz-Farny circle. It also lies on the Feuerbach hyperbola, Jerabek hyperbola, and Kiepert hyperbola, as well as the Darboux cubic, M'Cay cubic, Neuberg cubic, orthocubic, and Thomson cubic.

Distances to some named centers include

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)

where is the Clawson point, is the triangle centroid, is the Gergonne point, in is incenter, is the symmedian point, is the de Longchamps point, is the mittenpunkt, is the nine-point center, is the Nagel point, is the circumcenter, is the Spieker center, is the triangle area, is the circumradius, and is Conway triangle notation.

Relationships involving the orthocenter include the following:

			(16)
			(17)
			(18)
			(19)

where is the area, is the circumradius of the reference triangle, and , , , and is Conway triangle notation (P. Moses, pers. comm., Feb. 23, 2005). In the case of an acute triangle,

			(20)
			(21)

where is the inradius and

(22)

is the inradius of the orthic triangle (Johnson 1929, p. 191).

Another orthocenter relation is given by

(23)

where is the circumcenter.

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).

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References

Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. 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. "More on the Altitudes and Orthocenter of a Triangle." Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 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.Honsberger, R. "The Orthocenter." Ch. 2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 17-26, 1995.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." https://faculty.evansville.edu/ck6/tcenters/class/orthocn.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(4)=Orthocenter." https://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X4.Satterly, J. "2997. Relations between the Portions of the Altitudes of a Plane Triangle." Math. Gaz. 46, 50-51, 1962.Vandeghen, A. "Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle." Amer. Math. Monthly 72, 1091-1094, 1965.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 165, 1991.

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Orthocenter

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Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Orthocenter.html

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