Feuerbach hyperbola

It has the trilinear equation ${\displaystyle {\frac {\cos B-\cos C}{\alpha }}+{\frac {\cos C-\cos A}{\beta }}+{\frac {\cos A-\cos B}{\gamma }}}$(here ${\displaystyle A,B,C}$ are the angles at the respective vertices and ${\displaystyle (\alpha ,\beta ,\gamma )}$ is the barycentric coordinate).[2]

Like other rectangular hyperbolas, the orthocenter of any three points on the curve lies on the hyperbola. So, the orthocenter of the triangle ${\displaystyle ABC}$ lies on the curve.

The line ${\displaystyle OI}$ is tangent to this hyperbola at ${\displaystyle I}$.

Kariya's theorem

Given a triangle ${\displaystyle ABC}$, let ${\displaystyle A_{1},B_{1},C_{1}}$ be the touchpoints of the incircle ${\displaystyle \odot I}$ with the sides of the triangle opposite to vertices ${\displaystyle A,B,C}$ respectively. Let ${\displaystyle X,Y,Z}$ be points lying on the lines ${\displaystyle IA_{1},IB_{1},IC_{1}}$ such that ${\displaystyle IX=IY=IZ}$. Then, the lines ${\displaystyle AX,BY,CZ}$ are concurrent at a point lying on the Feuerbach hyperbola.

The Kariya's theorem has a long history.[4] It was proved independently by Auguste Boutin and V. Retali.,[5][6][7] but it became famous only after Kariya's paper.[8] Around that time, many generalizations of this result were given. Kariya's theorem can be used for the construction of the Feuerbach hyperbola.

Both Lemoine's theorem and Kariya's theorem are a special case of Jacobi's theorem.

• Feuerbach Hyperbola, MathWorld