Kiepert hyperbola

Kiepert hyperbola with triangular centers

A Kiepert triangle (red) with a perspective center

At the sides of a triangle ABC three similar isosceles triangles are added, each with one side of the given triangle as a base. Then the tips of the three isosceles triangles form a new triangle called the Kiepert triangle (after the German mathematician Ludwig Kiepert ).

The Kiepert hyperbola is the geometric location of all perspective centers of Kiepert triangles of triangle ABC. It is an equilateral hyperbola that goes through the following points, among others:

• the corners of the given triangle,
• the height intersection ,
• the focus ,
• the Spieker point ,
• the two Napoleon points ,
• the two Fermat points ,
• the two Vecten points .

Designations and coordinates

The base angle of the attached isosceles triangles is taken positive if they are directed outwards, otherwise negative. The associated Kiepert triangle is designated with , the center of perspective with . ${\displaystyle \phi }$ ${\displaystyle {\mathcal {K}}_{\phi }}$ ${\displaystyle K_{\phi }}$

Barycentric coordinates of (using Conway triangle notation ): ${\displaystyle K_{\phi }}$

${\displaystyle \left({\frac {1}{S_{A}+S_{\phi }}}:{\frac {1}{S_{C}+S_{\phi }}}:{\frac {1}{S_{C}+S_{\phi }}}\right)}$

The formula for the Kiepert hyperbola in barycentric coordinates is

${\displaystyle (b^{2}-c^{2})yz+(c^{2}-a^{2})xz+(a^{2}-b^{2})xy=0.\,}$

The center of the Kiepert hyperbola has the barycentric coordinates

${\displaystyle (b^{2}-c^{2})^{2}:(c^{2}-a^{2})^{2}:(a^{2}-b^{2})^{2},\,}$

the Kimberling number X (115) and is on the Feuerbach district (nine-point circle).

properties

• The Kiepert hyperbola is isogonally conjugated to the Brocard axis .

literature

• RH Eddy, R. Fritsch: The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle . Mathematics Magazine, Vol. 67, No. 3 (June, 1994), pp. 188-205

Web links

Commons : Kiepert's hyperbola  - collection of images, videos and audio files

• Eric W. Weisstein : Kiepert Hyperbola . In: MathWorld (English).