Johnson Circles - Properties

Properties

1. The centers of the Johnson circles lie on a circle of the same radius r as the Johnson circles centered at H. These centers form the Johnson triangle.
2. The circle centered at H with radius 2r, known as the anticomplementary circle is tangent to each of the Johnson circles. The three tangent points are reflections of point H about the vertices of the Johnson triangle.
3. The points of tangency between the Johnson circles and the anticomplementary circle form another triangle, called the anticomplementary triangle of the reference triangle. It is similar to the Johnson triangle, and is homothetic by a factor 2 centered at H, their common circumcenter.
4. Johnson's theorem: The 2-wise intersection points of the Johnson circles (vertices of the reference triangle ABC) lie on a circle of the same radius r as the Johnson circles. This property is also well known in Romania as The 5 lei coin problem of Gheoghe Ţiţeica.
5. The reference triangle is in fact congruent to the Johnson triangle, and is homothetic to it by a factor −1.
6. The point H is the orthocenter of the reference triangle and the circumcenter of the Johnson triangle.
7. The homothetic center of the Johnson triangle and the reference triangle is their common nine-point center.