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Kiepert Hyperbola


The Kiepert hyperbola is a hyperbola and triangle conic that is related to the solution of Lemoine's problem and its generalization to isosceles triangles constructed on the sides of a given triangle.

The vertices of the constructed triangles are given in trilinear coordinates by

A^'=-sinphi:sin(C+phi):sin(B+phi)
(1)
B^'=sin(C+phi):-sinphi:sin(A+phi)
(2)
C^'=sin(B+phi):sin(A+phi):-sinphi,
(3)

where phi is the base angle of the isosceles triangle.

Kiepert (1869) showed that the lines connecting the vertices of the given triangle and the corresponding peaks of the isosceles triangles concur. The trilinear coordinates of the point of concurrence are

 sin(B+phi)sin(C+phi):sin(C+phi)sin(A+phi):sin(A+phi)sin(B+phi).
(4)

The locus of this point as the base angle varies is given by the curve with trilinear coordinates

 (sin(B-C))/alpha+(sin(C-A))/beta+(sin(A-B))/gamma=0,
(5)

or equivalently

 (bc(b^2-c^2))/alpha+(ca(c^2-a^2))/beta+(ab(a^2-b^2))/gamma=0
(6)

(Kimberling 1998, p. 237). This curve is a rectangular hyperbola known as the Kiepert hyperbola.

KiepertHyperbola

Kiepert center is Kimberling center X_(115), which has equivalent triangle center functions

alpha_(115)=((b^2-c^2)^2)/a
(7)
alpha_(115)=asin^2(B-C)
(8)

(Kimberling 1998, p. 86).

The Kiepert hyperbola passes through Kimberling centers X_i for i=2, (triangle centroid), 4 (orthocenter), 10 (Spieker center; i.e., incenter of the medial triangle of DeltaABC; Eddy and Fritsch 1994), 13 (first Fermat point), 14 (second Fermat point), 17 (first Napoleon point), 18 (second Napoleon point), 76 (third Brocard point), 83 (isogonal conjugate of the Brocard midpoint; Eddy and Fritsch 1994), 94, 96, 98 (Tarry point), 226, 262, 275, 321, 485 (outer vecten point), 486 (inner vecten point), 598, 671, 801, 1029, 1131, 1132, 1139 (inner pentagon point), 1140 (outer pentagon point), 1327, 1328, 1446, 1676, 1677, 1751, 1916, 2009, 2010, 2051, 2052, 2394, 2592, 2593, 2671 (first golden arbelos point), 2672 (second golden arbelos point), 2986, and 2996

A subset of these points is summarized in the following table together with their corresponding angles phi (Eddy and Fritsch 1994, p. 193; Kimberling 1998, pp. 176-178 and 237). Here, omega is the Brocard angle.

KiepertHyperbolaMedialTriangle

Eddy and Fritsch (1994) also showed that the Kiepert hyperbola passes through the Spieker center.

The asymptotes of the Kiepert hyperbola are the Simson lines of the intersections of the Brocard axis with the circumcircle.

The isogonal conjugate of the Kiepert hyperbola is the Brocard axis, and the isotomic conjugate is the line GK through the triangle centroid G and the symmedian point K.

Writing the trilinear coordinates as

 alpha_i=d_is_i,
(9)

where d_i is the distance to the side opposite alpha_i of length s_i and using the point-line distance formula with (x_0,y_0) written as (x,y),

 d_i=(|((y_(i+2)-y_(i+1))(x-x_(i+1)))/(s_i)|-(x_(i+2)-x_(i+1))(y-y_(i+1)))/(s_i),
(10)

where y_4=y_1 and y_5=y_2 gives the formula

sum_(i=1)^(3)s_(i+1)s_(i+2)(s_(i+1)^2-s_(i+2)^2)(s_i)/((y_(i+2)-y_(i+1))(x-x_(i+1))-(x_(i+2)-x_(i+1))(y-y_(i+1)))=0
(11)
sum_(i=1)^(3)((s_(i+1)^2-s_(i+2)^2))/((y_(i+2)-y_(i+1))(x-x_(i+1))-(x_(i+2)-x_(i+1))(y-y_(i+1)))=0.
(12)

Bringing this equation over a common denominator then gives a quadratic in x and y, which is a conic section (in fact, a hyperbola). The curve can also be written as csc(A+t):csc(B+t):csc(C+t), as t varies over [-pi/4,pi/4].


See also

Brocard Angle, Brocard Axis, Brocard Points, Circumcircle, Fermat Points, Isogonal Conjugate, Isosceles Triangle, Kiepert Antipode, Kiepert Center, Kiepert Parabola, Lemoine's Problem, Nine-Point Circle, Orthocenter, Simson Line, Triangle Centroid

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References

Bottema, O. and van Hoorn, M. C. "Problem 664." Nieuw Arch. Wisk. 1, 79, 1983.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., pp. 442-448, 1893.Courcouf, W. J. "Back to Areals." Math. Gaz. 57, 46-51, 1973.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.Grinberg, D. and Myakishev, A. "A Generalization of the Kiepert Hyperbola." Forum Geom. 4, 253-260, 2004. http://forumgeom.fau.edu/FG2004volume4/FG200429index.html.Kelly, P. J. and Merriell, D. "Concentric Polygons." Amer. Math. Monthly 71, 37-41, 1964.Kiepert, L. "Solution de question 864." Nouv. Ann. Math. 8, 40-42, 1869.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lemoine, É. "Question 864." Nouv. Ann. Math. 7, 191, 1868.M'Cay, M. "Sur l'hyperbole de Kiepert." Mathesis 7, 208-220, 1887.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.Thébault, V. "On the Cevians of a Triangle." Amer. Math. Monthly 60, 167-173, 1953.van der Meiden, W. "Solution to Problem 664." Nieuw Arch. Wisk. 4, 385-286, 1983.van Lamoen, F. and Yiu, P. "The Kiepert Pencil of Kiepert Hyperbolas." Forum Geom. 1, 125-132, 2001. http://forumgeom.fau.edu/FG2001volume1/FG200118index.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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Kiepert Hyperbola

Cite this as:

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

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