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 
.
