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 , then the Johnson circles that are congruent to the circumcircle of have centers

			(1)
			(2)
			(3)

where , , , and are Conway triangle notation.

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

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

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

Portions of this entry contributed by Frank Jackson

CITE THIS AS:

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