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Orthocenter


Orthocenter

The intersection H of the three altitudes AH_A, BH_B, and CH_C of a triangle is called the orthocenter. 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

 cosBcosC:cosCcosA:cosAcosB.
(1)

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

 secA:secB:secC.
(2)

The orthocenter is Kimberling center X_4.

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

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

CircumcenterOrthocenter

The circumcenter O and orthocenter H 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

HCl=(8a^2b^2c^2|cosA||cosB||cosC|)/((a+b+c)(a^5-ba^4-ca^4+2bc^2a^2+2b^2ca^2-b^4a-c^4a+2b^2c^2a+b^5+c^5-bc^4-b^4c))
(3)
HG=2/3OH
(4)
HGe=(sqrt(a^(10)-b^2a^8-c^2a^8+b^2c^2a^6-b^8a^2-c^8a^2+b^2c^6a^2+b^6c^2a^2+b^(10)+c^(10)-b^2c^8-b^8c^2))/(4Delta^2(a^2+b^2+c^2))
(5)
HI=sqrt(2r^2+4R^2-S_omega)
(6)
HK=1/(4Delta(a^2+b^2+c^2))(sqrt(a^(10)-b^2a^8-c^2a^8+b^2c^2a^6-b^8a^2-c^8a^2+b^2c^6a^2+b^6c^2a^2+b^(10)+c^(10)-b^2c^8-b^8c^2))
(7)
HL=2OH
(8)
HM=((a^3-ba^2-ca^2-b^2a-c^2a-2bca+b^3+c^3-bc^2-b^2c)IL)/((a+b+c)(a^2-2ba-2ca+b^2+c^2-2bc))
(9)
HN=1/2OH
(10)
HNa=2OI
(11)
HO=(sqrt(a^6-b^2a^4-c^2a^4-b^4a^2-c^4a^2+3b^2c^2a^2+b^6+c^6-b^2c^4-b^4c^2))/(4Delta)
(12)
=sqrt(9R^2-(a^2+b^2+c^2))
(13)
=sqrt(9R^2-2S_omega)
(14)
HSp=1/2IL,
(15)

where Cl is the Clawson point, G is the triangle centroid, Ge is the Gergonne point, I in is incenter, K is the symmedian point, L is the de Longchamps point, M is the mittenpunkt, N is the nine-point center, Na is the Nagel point, O is the circumcenter, Sp is the Spieker center, Delta is the triangle area, R is the circumradius, and S_omega is Conway triangle notation.

Relationships involving the orthocenter include the following:

a^2+b^2+c^2+AH^2+BH^2+CH^2=(3a^2b^2c^2)/(4Delta^2)
(16)
=12R^2
(17)
AH+BH+CH=(abc)/(2Delta)(|cosA|+|cosB|+|cosC|)
(18)
AH^2+BH^2+CH^2=4R^2-(2S_AS_BS_C)/(S^2),
(19)

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

AH+BH+CH=2(r+R)
(20)
AH^2+BH^2+CH^2=4R(R-r_H),
(21)

where r is the inradius and

 r_H=2R|cosAcosBcosC|
(22)

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

Another orthocenter relation is given by

 AH^2+BH^2+CH^2=OH^2+3R^2,
(23)

where O 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).


See also

Circumcenter, Droz-Farny Circles, Euler Line, Fuhrmann Circle, Incenter, Orthic Triangle, Orthocentric Coordinates, Orthocentric Quadrilateral, Orthocentric System, Polar Circle, Triangle Centroid

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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.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.Honsberger, R. "The Orthocenter." Ch. 2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 17-26, 1995.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Orthocenter." http://faculty.evansville.edu/ck6/tcenters/class/orthocn.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(4)=Orthocenter." http://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: Penguin, p. 165, 1991.

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Orthocenter

Cite this as:

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

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