Nine-point hyperbola

Bôcher notes that when P is the orthocenter, one obtains the nine-point circle, and when P is on the circumcircle of △ABC, then the conic is an equilateral hyperbola.

An approach to the nine-point hyperbola using the analytic geometry of split-complex numbers was devised by E. F. Allen in 1941.

, j2 = 1, he uses split-complex arithmetic to express a hyperbola as It is used as the circumconic of triangle

They requisitioned the unit circle in the complex plane as the circumcircle of the given triangle.

[2] For Yaglom, a hyperbola is a Minkowskian circle as in the Minkowski plane.

[3] He considers a triangle inscribed in a "circumcircle" which is in fact a hyperbola.

In the Minkowski plane the nine-point hyperbola is also described as a circle: In 2005 J.

Christopher Bath[5] describes a nine-point rectangular hyperbola passing through these centers: incenter X(1), the three excenters, the centroid X(2), the de Longchamps point X(20), and the three points obtained by extending the triangle medians to twice their cevian length.

Points of triangle △ ABC and given point P
Six constituent lines of the quadrangle formed by A, B, C, P
Nine-point hyperbola . The right branch bisects BA , BC , BP ; the left bisects PA , PC , AC , and passes through the intersections of lines BC, PA and AB, PC .