Like other rectangular hyperbolas, the orthocenter of any three points on the curve lies on the hyperbola. So, the orthocenter of the triangle lies on the curve.
The line is tangent to this hyperbola at .
Isogonal conjugate of OI
The hyperbola is the isogonal conjugate of , 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.
Kariya's theorem
Given a triangle , let be the touchpoints of the incircle with the sides of the triangle opposite to vertices respectively. Let be points lying on the lines such that . Then, the lines 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.