Feuerbach hyperbola

In geometry, the Feuerbach hyperbola is a rectangular hyperbola passing through important triangle centers such as the Orthocenter, Gergonne point, Nagel point and Schiffler point.

The center of the hyperbola is the Feuerbach point, the point of tangency of the incircle and the nine-point circle.

are the angles at the respective vertices and

( α , β , γ )

, the line joining the circumcenter and the incenter.

[3] This fact leads to a few interesting properties.

Specifically all the points lying on the line

have their isogonal conjugates lying on the hyperbola.

The Nagel point lies on the curve since its isogonal conjugate is the point of concurrency of the lines joining the vertices and the opposite Mixtilinear incircle touchpoints, also the in-similitude of the incircle and the circumcircle.

Similarly, the Gergonne point lies on the curve since its isogonal conjugate is the ex-similitude of the incircle and the circumcircle.

The pedal circle of any point on the hyperbola passes through the Feuerbach point, the center of the hyperbola.

with the sides of the triangle opposite to vertices

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 [ja]'s paper.

Kariya's theorem can be used for the construction of the Feuerbach hyperbola.