Johnson circles

In the geometry is meant by the Johnson - circles of a triangle three circles with the same radius, which go through two corners and have one point in common. The triangle formed by the centers of these circles is called a Johnson - triangle. The name goes back to the U.S. surveyor Roger Arthur Johnson ( 1890-1954 ).

Properties

• The three Johnson - circles of a triangle exist and are uniquely determined.
• The three Johnson circles have the same radius as the radius of the given triangle.

• The Johnson Triangle and the given triangle are congruent. The center of rotation of the congruence is the center of Feuerbach circle.
• The perpendicular bisectors of the given triangle are the heights in the Johnson triangle, the altitudes of the given triangle is the perpendicular bisector of the Johnson Triangle.
• Therefore, the common point of the three Johnson circles the orthocenter of the given triangle and thus the circumcenter of the Johnson Triangle.
• Also why the orthocenter of the Johnson triangle is the circumcenter of the given triangle.