“Kiepert's Hyperbola”的概念、定义、习题解题方法及翻译-数学辞典

单词

Kiepert's Hyperbola

释义

Kiepert's Hyperbola

A curve which is related to the solution of Lemoine's Problem and its generalization to IsoscelesTriangles constructed on the sides of a given Triangle. The Vertices of the constructed Triangles are

			(1)
			(2)
			(3)

where

is the base Angle of the Isosceles Triangle. Kiepert showed that the lines connecting theVertices of the given Triangle and the corresponding peaks of the Isosceles Triangles Concur. The Trilinear Coordinates of the point of concurrence are

(4)

The locus of this point as the base Angle varies is given by the curve

(5)

Writing the Trilinear Coordinates as

(6)

where

is the distance to the side opposite

of length

and using the Point-Line Distance Formula with

written as

(7)

where

and

gives the Formula

(8)

(9)

Bringing this equation over a common Denominator then gives a quadratic in

and

, which is a Conic Section(in fact, a Hyperbola). The curve can also be written as

, as

varies over

Kiepert's hyperbola passes through the triangle's Centroid (), Orthocenter (), Vertices ( if and if), (), (), Fermat Point (), second IsogonicCenter (), Isogonal Conjugate of the Brocard Midpoint (), andBrocard's Third Point (), where is the Brocard Angle (Eddy andFritsch 1994, p. 193).

The Asymptotes of Kiepert's hyperbola are the Simson Lines of the intersections ofthe Brocard Axis with the Circumcircle. Kiepert's hyperbola is a Rectangular Hyperbola. In fact, allnondegenerate conics through the Vertices and Orthocenter of a Triangle areRectangular Hyperbolas the centers of which lie halfway between the Isogonic Centersand on the Nine-Point Circle. The Locus of centers of these Hyperbolas is theNine-Point Circle.

The Isogonal Conjugate curve of Kiepert's hyperbola is the Brocard Axis. The center of the Incircleof the Triangle constructed from the Midpoints of the sides of a given Triangle lies onKiepert's hyperbola of the original Triangle.

See also Brocard Angle, Brocard Axis, Brocard Points, Centroid (Triangle), Circumcircle,Isogonal Conjugate, Isogonic Centers, Isosceles Triangle, Lemoine's Problem,Nine-Point Circle, Orthocenter, Simson Line

References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893.

Eddy, R. H. and Fritsch, R. ``The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle.'' Math. Mag. 67, 188-205, 1994.

Kelly, P. J. and Merriell, D. ``Concentric Polygons.'' Amer. Math. Monthly 71, 37-41, 1964.

Mineuer, A. ``Sur les asymptotes de l'hyperbole de Kiepert.'' Mathesis 49, 30-33, 1935.

Rigby, J. F. ``A Concentrated Dose of Old-Fashioned Geometry.'' Math. Gaz. 57, 296-298, 1953.

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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