Orthocentric system

In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three.

The center of this common nine-point circle lies at the centroid of the four orthocentric points.

The common nine-point circle is tangent to all 16 incircles and excircles of the four triangles whose vertices form the orthocentric system.

Join each of the four orthocentric points to their common nine-point center and extend them into four lines.

If a point P is chosen on the Euler line HN of the reference triangle △ABC with a position vector p such that p = n + α(h – n) where α is a pure constant independent of the positioning of the four orthocentric points and three more points PA, PB, PC such that pa = n + α(a – n) etc., then P, PA, PB, PC form an orthocentric system.

Consequently the circumcircles of the four triangles △ABC, △ABH, △ACH, △BCH are all equal and form a set of Johnson circles as shown in the diagram adjacent.

These equations together with the law of sines result in the identity Feuerbach's theorem states that the nine-point circle is tangent to the incircle and the three excircles of a reference triangle.

The locus of the perspectors of this family of rectangular hyperbolas will always lie on the four orthic axes.

So if a rectangular hyperbola is drawn through four orthocentric points it will have one fixed center on the common nine-point circle but it will have four perspectors one on each of the orthic axes of the four possible triangles.

The well documented rectangular hyperbolas that pass through four orthocentric points are the Feuerbach, Jeřábek and Kiepert circumhyperbolas of the reference triangle △ABC in a normalized system with H as the orthocenter.

Orthocentric system . Any point is the orthocenter of the triangle formed by the other three.
Common nine-point circle , where N, O 4 , A 4 are the nine-point center, circumcenter, and orthocenter respectively of the triangle formed from the other three orthocentric points A 1 , A 2 , A 3 .
Orthocentric system . Where O 1 , O 2 , O 3 , O 4 are the circumcenters of the four possible triangles formed from the orthocentric points A 1 , A 2 , A 3 , A 4 .