Kiepert Parabola

Kiepert parabolaKiepertsParabolaFocus

Let three similar isosceles triangles DeltaA^'BC, DeltaAB^'C, and DeltaABC^' be constructed on the sides of a triangle DeltaABC. Then DeltaABC and DeltaA^'B^'C^' are perspective triangles, and the envelope of their perspectrix as the vertex angle of the erected triangles is varied is a parabola known as the Kiepert parabola. It has trilinear conic function


This parabola was first studied by Artzt (1884; Eddy and Fritsch 1994).


The Euler line of a triangle is the conic section directrix of the Kiepert parabola. In fact, the directrices of all parabolas inscribed in a triangle pass through the orthocenter. The Brianchon point for the Kiepert parabola is the Steiner point S of the reference triangle, and the triangle DeltaS_AS_BS_C formed by the points of contact is called the Steiner triangle.

The Kiepert parabola is tangent to the sides of the triangle (or their extensions), the line at infinity, and the Lemoine axis. The focus of the parabola has triangle center function


and is Kimberling center X_(110).

The Kiepert parabola passes through Kimberling centers X_i for i=523 (the isogonal conjugate of the focus of the Kiepert parabola X_(110)), 669 (the crossdifference of X_2 and X_(39), 1649, and 2528 (Weisstein, Oct. 16 and Dec. 13, 2004).

The polar triangle of the Kiepert parabola is the Steiner triangle.


The Kiepert parabola focus and Parry point are the two intersections of a triangle's circumcircle with its Parry circle.


X_(110) is also the Feuerbach point of the tangential triangle of DeltaABC.

See also

Brianchon Point, Envelope, Euler Line, Isosceles Triangle, Kiepert Hyperbola, Lemoine Axis, Parabola, Parry Circle, Parry Point, Steiner Points, Steiner Triangle

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Artzt, A. "Untersuchungen über ähnliche Punktreihen auf den Seiten eines Dreiecks und auf deren Mittelsenkrechten, sowie über kongruente Strahlenbüschel aus den Ecken desselben; ein Beitrag zur Geometrie des Brocardschen Kreises." Programm des Gymnasiums zu Recklinghausen 54, 3-22, 1884,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.Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 26, 1913.Kimberling, C. "Encyclopedia of Triangle Centers: X(110)=Focus of Kiepert Parabola.", J. B. J. "Sur la parabole de Kiepert." Ann. de la Soc. scientifique de Bruxelles, 1-11, 1909-1910.Neuberg, J. B. J. "Über die Kiepertsche Parabel." Mitteilungen der naturforsch. Gessel. in Bern, 1-11, 1911.

Referenced on Wolfram|Alpha

Kiepert Parabola

Cite this as:

Weisstein, Eric W. "Kiepert Parabola." From MathWorld--A Wolfram Web Resource.

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