Let three equal circles with centers 
, 
, and 
intersect in a single point
and intersect
pairwise in the points 
,
, and 
. Then the circumcircle 
of the reference
triangle 
is congruent to the original three.
Furthermore, the points 
, 
, 
, and 
form an orthocentric system.
Here, the original three circles are known as Johnson circles and the triangle 
formed by their centers is known as the Johnson
triangle. Amazingly, the Johnson
triangle circumcircle is also congruent to the circumcircle
of the reference triangle and centered at the
orthocenter 
.
A "triquetra" is a figure consisting of three circular arcs of equal radius, and has seen extensive use in heraldry (i.e., coats of arms), specifically in the case of the so-called Borromean rings. The term "Triquetra theorem" was used by Mackenzie (1992) to describe Johnson's theorem.
Mackenzie (1992) generalized this theorem to the case where the three circles do not coincide. In this case, they form six intersection points, and if you partition the points into any two groups of three and look at the circumradii of the points in those groups, there is a nice formula relating them to the radii of the triquetra circles. This formula has some pretty geometric consequences (or "porisms"). Ultimately, Johnson's theorem turns out to be closely related to Poncelet's porism.
