Kiepert conics

Summary

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 , and , constructed on the sides of a triangle as bases, are similar, isosceles and similarly situated, then the triangles and are in perspective. As the base angle of the isosceles triangles varies between and , the locus of the center of perspectivity of the triangles and 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 ...."

Kiepert hyperbola

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The Kiepert hyperbola was discovered by Ludvig Kiepert while investigating the solution of the following problem proposed by Emile Lemoine in 1868: "Construct a triangle, given the peaks of the equilateral triangles constructed on the sides." A solution to the problem was published by Ludvig Kiepert in 1869 and the solution contained a remark which effectively stated the locus definition of the Kiepert hyperbola alluded to earlier.[2]

Basic facts

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Let   be the side lengths and   the vertex angles of the reference triangle  .

Equation

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The equation of the Kiepert hyperbola in barycentric coordinates   is

 

Center, asymptotes

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  • The centre of the Kiepert hyperbola is the triangle center X(115). The barycentric coordinates of the center are
 .

Properties

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Property 4, simulation with K moved on the Kiepert hyperbola and P moved on the FK, F=X(14)-the first Fermat point
  1. The center of the Kiepert hyperbola lies on the nine-point circle. The center is the midpoint of the line segment joining the isogonic centers of triangle   which are the triangle centers X(13) and X(14) in the Encyclopedia of Triangle Centers.
  2. The image of the Kiepert hyperbola under the isogonal transformation is the Brocard axis of triangle   which is the line joining the symmedian point and the circumcenter.
  3. Let   be a point in the plane of a nonequilateral triangle   and let   be the trilinear polar of   with respect to  . The locus of the points   such that   is perpendicular to the Euler line of   is the Kiepert hyperbola.
  4. Let ABC be a triangle with F is the first (or second) Fermat point, let K be arbitrary point on the Kiepert hyperbola. Let P be arbitrary point on line FK. The line through P and perpendicular to BC meets AK at A0. Define B0, C0 cyclically, then A0B0C0 is an equilateral triangle.[3]

Kiepert parabola

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The Kiepert parabola was first studied in 1888 by a German mathematics teacher Augustus Artzt in a "school program".[2][4]

Basic facts

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  • The equation of the Kiepert parabola in barycentric coordinates   is
 
where
 .
  • The focus of the Kiepert parabola is the triangle center X(110). The barycentric coordinates of the focus are
 
  • The directrix of the Kiepert parabola is the Euler line of triangle  .

Images

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See also

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  • Weisstein, Eric W. "Kiepert Hyperbola". MathWorld--A Wolfram Web Resource. Retrieved 5 February 2022.
  • Weisstein, Eric W. "Kiepert Parabola". MathWorld--A Wolfram Web Resource. Retrieved 5 February 2022.

References

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  1. ^ Kimberling, C. "X(110)=Focus of Kiepert Parabola". Encyclopedia of Triangle Centers. Retrieved 4 February 2022.
  2. ^ a b c Eddy, R. H.; Fritsch, R. (1994). "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle". Math. Mag. 67 (3): 188–205. doi:10.1080/0025570X.1994.11996212.
  3. ^ Dao Thanh Oai (2018), "Some new equilateral triangles in a plane geometry." Global J Adv Res Classical Mod Geometries Vol 7, Isue 2, pages 73-91.
  4. ^ Sharp, J. (2015). "Artzt parabolas of a triangle". The Mathematical Gazette. 99 (546): 444–463. doi:10.1017/mag.2015.81. S2CID 123814409.