Johnson Circumconic


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


and so has trilinear equation


It passes through Kimberling centers X_(110) (focus of the Kiepert parabola), X_(265) (reflection of the circumcenter X_3 in the Jerabek antipode X_(125)), and X_(1625).

It has center at the nine-point center X_5.

Its area is


Interestingly, the point X_(110) is the intersection of the Johnson circumconic and both the circumcircle of the reference triangle and the MacBeath circumconic. The point X_(265) 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 X_5 (F. M. Jackson, pers. comm., Mar. 15, 2006).

The Johnson circumconic meets the Steiner circumellipse at a point with center function


which is the isotomic conjugate of X_(520). It is also the intersection of lines X_4X_(290), X_5X_(276), X_(99)X_(107), X_(190)X_(823), X_(264)X_(339), X_(664)X_(811), and X_(670)X_(877), as well as the trilinear pole of the line through X_i for i=2, 216, 232, 264, 324, 393, 2052, and 2404 (P. Moses, pers. comm., Mar. 22, 2006).

Let S, S_A, S_B, and S_C be Conway triangle notation. For a point p:q:r on the circumcircle, the point with


is on the Johnson circumconic. For a point p:q:r on the Steiner circumellipse, the point with


is on the Johnson circumconic. For a point p:q:r on the MacBeath circumconic, the point with


is on the Johnson circumconic. For a point p:q:r on the line at infinity, the point with


is on the Johnson circumconic (P. Moses, pers. comm., Mar. 22, 2006).

See also

Circumconic, Johnson Triangle, Johnson Triangle Circumcircle

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Johnson Circumconic." From MathWorld--A Wolfram Web Resource.

Subject classifications