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 ninepoint 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 ninepoint center (F. M. Jackson, pers. comm., Mar. 15, 2006).
The Johnson circumconic meets the Steiner circumellipse at a point with center function
(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).