The Johnson Circles

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The Johnson circles are a triplet of congruent circles sharing a single point. Every triangle has exactly two Johnson triplets.

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Properties:

• The locators are the centers of the three circles. They form the Johnson triangle with circumcircle of the same radius.

• Johnson's theorem: the "reference triangle" with vertices the points of two-fold intersection has, surprisingly, a circumcircle of the same radius.

• The reference triangle is congruent to the Johnson triangle by homothety of factor .

• The anticomplementary circle with twice the radius touches the Johnson circles.

• The inscribed anticomplementary triangle is homothetic to the Johnson triangle with factor 2.

• The three locators and the origin are, surprisingly, such that each is the orthocenter of the three others.

• The homothetic center of the Johnson and reference triangle is the center of the nine-point circle of the reference triangle.

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Contributed by: Claude Fabre (March 2011)
Open content licensed under CC BY-NC-SA


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