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The Johnson Circles
The Johnson circles are a triplet of congruent circles sharing a single point. Every triangle has exactly two Johnson triplets.
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.
Contributed by:
Claude Fabre
THINGS TO TRY
Drag Locators
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RELATED LINKS
Circle
(
Wolfram
MathWorld
)
Johnson Circles
(
Wolfram
MathWorld
)
Yff Circles
(
Wolfram
MathWorld
)
Johnson-Yff Circles
(
Wolfram
MathWorld
)
Orthocentric System
(
Wolfram
MathWorld
)
PERMANENT CITATION
"
The Johnson Circles
" from
the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/TheJohnsonCircles/
Contributed by:
Claude Fabre
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