Orthocentric System - Further Properties

Further Properties

The four Euler lines of an orthocentric system are orthogonal to the four orthic axes of an orthocentric system.

The six connectors that join any pair of the original four orthocentric points will produce pairs of connectors that are orthogonal to each other such that they satisfy the distance equations

where R is the common circumradius of the four possible triangles. 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. Because the nine-point circle is common to all four possible triangles in an orthocentric system it is tangent to 16 circles comprising the incircles and excircles of the four possible triangles.

Any conic that passes through the four orthocentric points can only be a rectangular hyperbola. This is a result of Feuerbach's conic theorem that states that for all circumconics of a reference triangle that also passes through its orthocenter, the locus of the center of such circumconics form the nine-point circle and that the circumconics can only be rectangular hyperbolas.

Note that 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. Note also that the one point on the nine-point circle that is the center of this rectangular hyperbola will have four different definitions dependent on which of the four possible triangles is used as the reference triangle.

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.

The four possible triangles have a set of four inconics known as the orthic inconics that share certain properties. The contacts of these inconics with the four possible triangles occur at the vertices of their common orthic triangle. In a normalized orthocentric system the orthic inconic that is tangent to the sides of the triangle ABC is an inellipse and the orthic inconics of the other three possible triangles are hyperbolas. These four orthic inconics also share the same Brianchon point, H, the orthocentric point closest to the common nine-point center. The centers of these orthic inconics are the symmedian points, K of the four possible triangles.

There are many documented cubics that pass through a reference triangle and its orthocenter. The circumcubic known as the orthocubic - K006 is interesting in that it passes through three orthocentric systems as well as the three vertices of the orthic triangle (but not the orthocenter of the orthic triangle). The three orthocentric systems are the incenter and excenters, the reference triangle and its orthocenter and finally the orthocenter of the reference triangle together with the three other intersection points that this cubic has with the circumcircle of the reference triangle.