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Johnson Circles


JohnsonCircles

Johnson's theorem states that if three equal circles mutually intersect one another in a single point, then the circle passing through their other three pairwise points of intersection is congruent to the original three circles. If the pairwise intersections are taken as the vertices of a reference triangle DeltaABC, then the Johnson circles that are congruent to the circumcircle of DeltaABC have centers

J_A=-abcS_A:c(S^2+S_A+S_C):b(S^2+S_AS_B)
(1)
J_B=c(S^2+S_BS_C):-abcS_B:a(S^2+S_AS_B)
(2)
J_C=b(S^2+S_BS_C):a(S^2+S_AS_C):-abcS_C,
(3)

where S, S_A, S_B, and S_C are Conway triangle notation.

The centers of the Johnson circles form the Johnson triangle DeltaJ_AJ_BJ_C which, together with O, form an orthocentric system.

JohnsonCirclesConcurrence

The point of threefold concurrence of the Johnson circles is the orthocenter H of the reference triangle DeltaABC.

JohnsonCirclesCircumcircle

Note also that intersections of the directed lines from the orthocenter H of the reference triangle through the centers of the Johnson circles intersect the Johnson circles at the vertices of the anticomplementary triangle DeltaP_AP_BP_C. The anticomplementary circle, with center H and radius 2r (where r is the radius of the Johnson circles) therefore circumscribes the Johnson circles and is tangent to them at the points J_A, J_B, and J_C.


See also

Anticomplementary Circle, Anticomplementary Triangle, Johnson Midpoint, Johnson's Theorem, Johnson Triangle, Johnson-Yff Circles, Yff Circles

Portions of this entry contributed by Frank Jackson

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Cite this as:

Jackson, Frank and Weisstein, Eric W. "Johnson Circles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JohnsonCircles.html

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