Johnson Circle

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Triangle with Johnson circles (red) and Johnson triangle (blue)

In geometry , the Johnson circles of a triangle are three circles with the same radius, each of which goes through two corners and has one point in common. The triangle formed by the centers of these circles is called the Johnson triangle . The name goes back to the American geometer 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 perimeter of the given triangle.
  • The Johnson triangle and the given triangle are congruent. The center of rotation of the congruence mapping is the center of the Feuerbach circle .
  • The perpendiculars of the given triangle are the heights in the Johnson triangle, the heights of the given triangle are the perpendiculars of the Johnson triangle.
  • Therefore, the common point of the three Johnson circles is the vertical intersection of the given triangle and thus the circumcenter of the Johnson triangle.
  • This is also why the height intersection of the Johnson triangle is the circumcenter of the given triangle.

Web links

Commons : Johnson circles  - collection of images, videos and audio files

Individual evidence

  1. ^ Clark Kimberling: Roger Arthur Johnson (1890-1954), geometer. University of Evansville, August 22, 2002, accessed October 16, 2013 .