“Orthocenter”的概念、定义、习题解题方法及翻译-数学辞典

单词

Orthocenter

释义

Orthocenter

The intersection of the three Altitudes of a Triangleis 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 RightTriangle, 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 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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