The Johnson triangle 
,
a term coined here for the first time, is the triangle formed by the centers of the
Johnson circles.
It has trilinear vertex matrix
![[-abcS_A:c(S^2+S_AS_C):b(S^2+S_AS_B); c(S^2+S_BS_C):-abcS_B:a(S^2+S_AS_B); b(S^2+S_BS_C):a(S^2+S_AS_C):-abcS_C],](/images/equations/JohnsonTriangle/NumberedEquation1.svg) |
where 
,
, 
, and 
is Conway triangle
notation.
The Johnson triangle circumcircle is congruent to the Johnson circles and therefore also to the circumcircle of the reference triangle.
The Johnson triangle is congruent to the reference triangle, with which it is also in perspective with perspector at nine-point
center. Because the Johnson triangle is congruent to the reference triangle,
the points 
,
, 
, and 
form an orthocentric
system. Furthermore, the Johnson triangle is homothetic to the reference
triangle and has a homothetic center at the nine-point
center of the reference triangle (midway
between 
and 
on their common Euler line).
The nine-point center is also the midpoint of the lines 
,
, and 
. In fact, more generally, the nine-point
center lies at the center of a line between any defined point 
in the reference triangle and its congruent point 
in the Johnson triangle. (But note that this midpoint should
not be confused with the Johnson midpoint).
The radical lines through pairs of intersecting Johnson circles orthogonally bisect the sides of the Johnson triangle at its medians. Consequently, the medial triangle of the Johnson triangle is homothetic to the reference triangle and its homothetic center is the orthocenter of the reference triangle.
In addition to the vertices 
, 
, and 
being the reflections of 
in the sidelines 
, 
, and 
of 
, 
is also the Euler
triangle of the anticomplementary triangle 
is 
rotated by 
around 
, so 
and 
share the same nine-point
circle.
