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. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and the four circles passing through any three of the four points have the same radius.
If four points form an orthocentric system, then each of the four points is the orthocenter of the other three. These four possible triangles will all have the same nine-point circle. Consequently these four possible triangles must all have circumcircles with the same circumradius.
The common nine-point circle
The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the common nine-point circle passes. The nine-point circle also passes through the three orthogonal intersections at the feet of the altitudes of the four possible triangles.
This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three 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.
The common orthic triangle, its incenter, and its excenters
If the six connectors that join any pair of orthocentric points are extended to six lines that intersect each other, they generate seven intersection points. Four of these points are the original orthocentric points and the additional three points are the orthogonal intersections at the feet of the altitudes. The joining of these three orthogonal points into a triangle generates an orthic triangle that is common to all the four possible triangles formed from the four orthocentric points taken three at a time.
The incenter of this common orthic triangle must be one of the original four orthocentric points. Furthermore, the three remaining points become the excenters of this common orthic triangle. The orthocentric point that becomes the incenter of the orthic triangle is that orthocentric point closest to the common nine-point center. This relationship between the orthic triangle and the original four orthocentric points leads directly to the fact that the incenter and excenters of a reference triangle form an orthocentric system.
It is normal to distinguish one of the orthocentric points from the others, specifically the one that is the incenter of the orthic triangle; this one is denoted as the orthocenter of the outer three orthocentric points that are chosen as a reference triangle . In this normalized configuration, the point will always lie within the triangle , and all the angles of triangle will be acute. The four possible triangles referred above are then triangles . The six connectors referred above are . The seven intersections referred above are (the original orthocentric points), and (the feet of the altitudes of triangle and the vertices of the orthic triangle).
The orthocentric system and its orthic axes
The orthic axis associated with a normalized orthocentric system , where is the reference triangle, is a line that passes through three intersection points formed when each side of the orthic triangle meets each side of the reference triangle. Now consider the three other possible triangles, . They each have their own orthic axis.
Euler lines and homothetic orthocentric systems
Let vectors determine the position of each of the four orthocentric points and let be the position vector of , the common nine-point center. Join each of the four orthocentric points to their common nine-point center and extend them into four lines. These four lines now represent the Euler lines of the four possible triangles where the extended line is the Euler line of triangle and the extended line is the Euler line of triangle etc. If a point is chosen on the Euler line of the reference triangle with a position vector such that where is a pure constant independent of the positioning of the four orthocentric points and three more points such that etc., then form an orthocentric system. This generated orthocentric system is always homothetic to the original system of four points with the common nine-point center as the homothetic center and α the ratio of similitude.