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.
The vertices of the Johnson triangle, the reference triangle, and their two medial triangles, together with the circumcenter, orthocenter, and nine-point center, lie on the cubic curve catalogued as K026 by Gibert. This cubic curve is centrally symmetric about the nine-point center (Gibert; F. M. Jackson, pers. comm., Feb. 24, 2007).
