Kiepert conics information

In triangle geometry, the Kiepert conics are two special conics associated with the reference triangle. One of them is a hyperbola, called the Kiepert hyperbola and the other is a parabola, called the Kiepert parabola. The Kiepert conics are defined as follows:

If the three triangles ${\displaystyle A^{\prime }BC}$, ${\displaystyle AB^{\prime }C}$ and ${\displaystyle ABC^{\prime }}$, constructed on the sides of a triangle ${\displaystyle ABC}$ as bases, are similar, isosceles and similarly situated, then the triangles ${\displaystyle ABC}$ and ${\displaystyle A^{\prime }B^{\prime }C^{\prime }}$ are in perspective. As the base angle of the isosceles triangles varies between ${\displaystyle -\pi /2}$ and ${\displaystyle \pi /2}$, the locus of the center of perspectivity of the triangles ${\displaystyle ABC}$ and ${\displaystyle A^{\prime }B^{\prime }C^{\prime }}$ is a hyperbola called the Kiepert hyperbola and the envelope of their axis of perspectivity is a parabola called the Kiepert parabola.

It has been proved that the Kiepert hyperbola is the hyperbola passing through the vertices, the centroid and the orthocenter of the reference triangle and the Kiepert parabola is the parabola inscribed in the reference triangle having the Euler line as directrix and the triangle center X₁₁₀ as focus.^[1] The following quote from a paper by R. H. Eddy and R. Fritsch is enough testimony to establish the importance of the Kiepert conics in the study of triangle geometry:^[2]

"If a visitor from Mars desired to learn the geometry of the triangle but could stay in the earth's relatively dense atmosphere only long enough for a single lesson, earthling mathematicians would, no doubt, be hard-pressed to meet this request. In this paper, we believe that we have an optimum solution to the problem. The Kiepert conics ...."