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# Kiepert Parabola

Let three similar isosceles triangles , , and be constructed on the sides of a triangle . Then and 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 of the reference triangle, and the triangle 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 .

The Kiepert parabola passes through Kimberling centers for (the isogonal conjugate of the focus of the Kiepert parabola ), 669 (the crossdifference of and , 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.

is also the Feuerbach point of the tangential triangle of .

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

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## References

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." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X110.Neuberg, 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.

Kiepert Parabola

## Cite this as:

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