Johnson Circles - Further Properties

Further Properties

The three Johnson circles can be considered the reflections of the circumcircle of the reference triangle about each of the three sides of the reference triangle. Furthermore, under the reflections about the three sides of the reference triangle, its orthocenter H maps onto certain points on the circumcircle of the reference triangle that form the vertices of the circum-orthic triangle, and its circumcenter maps onto the vertices of the Johnson triangle.

The Johnson triangle and its reference triangle share the same nine-point center, the same Euler line and the same nine-point circle. The six points formed from the vertices of the reference triangle and its Johnson triangle all lie on the Johnson circumconic that is centred at the nine-point center and that has the point of the reference triangle as its perspector. The circumconic and the circumcircle share a fourth point, of the reference triangle.

Finally there are two interesting and documented circumcubics that pass through the six vertices of the reference triangle and its Johnson triangle as well as the circumcenter, the orthocenter and the nine-point center. The first is known as the first Musselman cubic – . This cubic also passes through the six vertices of the medial triangle and the medial triangle of the Johnson triangle. The second cubic is known as the Euler central cubic – . This cubic also passes through the six vertices of the orthic triangle and the orthic triangle of the Johnson triangle.

The X(i) point notation is the Clark Kimberling ETC classification of triangle centers.