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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 .