Orthocentric System - Euler Lines and Homothetic Orthocentric Systems

Euler Lines and Homothetic Orthocentric Systems

Let vectors a, b, c and h determine the position of each of the four orthocentric points and let n = (a + b + c + h) / 4 be the position vector of N, 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 HN is the Euler line of triangle ABC and the extended line AN is the Euler line of triangle BCH etc. 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 P_A, P_B, P_C such that p_a = n + α (a − n) etc., then P, P_A, P_B, P_C form an orthocentric system. This generated othocentric 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.

When P is chosen as the centroid G, then α = −1/3. When P is chosen as the circumcenter O, then α = −1 and the generated orthocentric system is congruent to the original system as well as being a reflection of it about the nine-point center. In this configuration P_A, P_B, P_C form a Johnson triangle of the original reference triangle ABC. 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.