Orthocentric System - The Common Orthic Triangle, Its Incenter and Excenters

The Common Orthic Triangle, Its Incenter and 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.

Note that 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 choose the orthocentric point that is the incenter of the orthic triangle as H the orthocenter of the outer three orthocentric points that are chosen as a reference triangle ABC. In this normalized configuration the point H will always lie within the triangle ABC and all the angles of triangle ABC will be acute. The four possible triangles referred above are then triangles ABC, ABH, ACH and BCH. The six connectors referred above are AB, AC, BC, AH, BH and CH. The seven intersections referred above are A, B, C, H (the original orthocentric points) and H_A, H_B, H_C (the feet of the altitudes of triangle ABC and the vertices of the orthic triangle).