Kiepert conics

Let ${\displaystyle a,b,c}$ be the side lengths and ${\displaystyle A,B,C}$ the vertex angles of the reference triangle ${\displaystyle ABC}$.

The equation of the Kiepert hyperbola in barycentric coordinates ${\displaystyle x:y:z}$ is

${\displaystyle {\frac {b^{2}-c^{2}}{x}}+{\frac {c^{2}-a^{2}}{y}}+{\frac {a^{2}-b^{2}}{z}}=0.}$

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

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

• The asymptotes of the Kiepert hyperbola are the Simson lines of the intersections of the Brocard axis with the circumcircle.
• The Kiepert hyperbola is a rectangular hyperbola and hence its eccentricity is ${\displaystyle {\sqrt {2}}}$.

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 ${\displaystyle ABC}$ 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 ${\displaystyle ABC}$ which is the line joining the symmedian point and the circumcenter.
3. Let ${\displaystyle P}$ be a point in the plane of a nonequilateral triangle ${\displaystyle ABC}$ and let ${\displaystyle p}$ be the trilinear polar of ${\displaystyle P}$ with respect to ${\displaystyle ABC}$. The locus of the points ${\displaystyle P}$ such that ${\displaystyle p}$ is perpendicular to the Euler line of ${\displaystyle ABC}$ is the Kiepert hyperbola.

• The equation of the Kiepert parabola in barycentric coordinates ${\displaystyle x:y:z}$ is

${\displaystyle f^{2}x^{2}+g^{2}y^{2}+h^{2}z^{2}-2fgxy-2ghyz-2hfzx=0}$
where
${\displaystyle f=(b^{2}-c^{2})/a,g=(c^{2}-a^{2})/b,h=(a^{2}-b^{2})/c}$.

• The focus of the Kiepert parabola is the triangle center X(110). The barycentric coordinates of the focus are

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

• The directrix of the Kiepert parabola is the Euler line of triangle ${\displaystyle ABC}$.