The Johnson circumconic, a term used here for the first time, is the circumconic that passes through the vertices of both the reference triangle and the Johnson triangle. It is a circumellipse for acute triangles and a circumhyperbola for obtuse triangles.
It has circumconic parameters
 |
(1)
|
and so has trilinear equation
 |
(2)
|
It passes through Kimberling centers 
(focus of the Kiepert
parabola), 
(reflection of the circumcenter 
in the Jerabek antipode 
), and 
.
It has center at the nine-point center 
.
Its area is
 |
(3)
|
Interestingly, the point 
is the intersection of the Johnson circumconic and both
the circumcircle of the reference
triangle and the MacBeath circumconic.
The point 
is the intersection of the Johnson circumconic with the Johnson
triangle circumcircle. Furthermore, these points are reflections of one another
in the nine-point center 
(F. M. Jackson, pers. comm., Mar. 15, 2006).
The Johnson circumconic meets the Steiner circumellipse at a point with center function
![alpha=1/(a[a^2(b^2-c^2)S_A^2]),](/images/equations/JohnsonCircumconic/NumberedEquation4.svg) |
(4)
|
which is the isotomic conjugate of 
. It is also the intersection of lines 
, 
, 
, 
, 
, 
, and 
, as well as the trilinear
pole of the line through 
for 
, 216, 232, 264, 324, 393, 2052, and 2404 (P. Moses,
pers. comm., Mar. 22, 2006).
Let 
,
,
,
and 
be Conway triangle notation. For a point
on the circumcircle, the point with
 |
(5)
|
is on the Johnson circumconic. For a point 
on the Steiner circumellipse,
the point with
 |
(6)
|
is on the Johnson circumconic. For a point 
on the MacBeath circumconic,
the point with
 |
(7)
|
is on the Johnson circumconic. For a point 
on the line at infinity,
the point with
 |
(8)
|
is on the Johnson circumconic (P. Moses, pers. comm., Mar. 22, 2006).
