The Thomson cubic
of a triangle
is the locus the centers of circumconics whose normals at the vertices are concurrent.
It is a self-isogonal cubic with pivot point
at the triangle centroid, so its parameter is
and its trilinear equation is given
by
(Cundy and Parry 1995; Kimberling 1998, p. 240).
It is sometimes called the seventeen-point cubic (Casey 1893, p. 460; Kimberling 1998, p. 240) because it passes through the vertices , ,
, the side midpoints , ,
, the altitude
midpoints ,
, , the excenters , , , the incenter (),
triangle centroid (),
circumcenter (),
orthocenter (),
and symmedian point ().
It also passes through the mittenpunkt (), as well as Kimberling
centers,
, and (Kimberling 1998, p. 240), as well as and so it is really a 23-point cubic!
of a triangle 
is the locus the centers of circumconics whose normals at the vertices are concurrent.
It is a self-isogonal cubic with pivot point
at the triangle centroid, so its parameter is
and its trilinear equation is given
by
, 
,
, the side midpoints 
, 
,
, the altitude
midpoints 
,
, 
, the excenters 
, 
, 
, the incenter 
(
),
triangle centroid 
(
),
circumcenter 
(
),
orthocenter 
(
),
and symmedian point 
(
).
It also passes through the mittenpunkt (
), as well as Kimberling
centers 
,
, and 
(Kimberling 1998, p. 240), as well as 
and 
so it is really a 23-point cubic!
