The Neuberg cubic 
of a triangle 
is the locus of all points 
whose reflections in the sidelines 
, 
,
and 
form a triangle perspective to 
. It is a self-isogonal
cubic with pivot point at the Euler infinity
point 
,
so it has parameter 
and trilinear equation
 |
(Cundy and Parry 1995).
It passes through Kimberling centers 
for 
(incenter 
), 3 (circumcenter 
), 4 (orthocenter 
), 13 (first Fermat point 
), 14 (second
Fermat point 
),
15 (first isodynamic point 
), 16 (second isodynamic
point 
),
30 (Euler infinity point), and 74 (Kimberling
1998, p. 240), in addition to 399 (Parry reflection point), 484 (first Evans
perspector), 616, 617, 1138, 1157, 1263, 1276 (second Evans perspector), 1277 (third
Evans perspector), 2118, 2132, and 2133. It also passes through excenters 
, 
, and 
of the reference triangle
and the circular points at infinity.
Let 
be a triangulation point and let 
be the circumcenter of
, 
the circumcenter of 
and 
the circumcenter of 
. Then lines 
, 
and 
are concurrent (at a point on the line joining the circumcenter
of 
and 
) if and only if 
lies on the union of the circumcircle
and the Neuberg cubic (Neuberg 1884).
Let 
be the Euler line of 
, 
the Euler line of 
and 
the Euler line of 
. The lines 
, 
and 
are concurrent (at a point on the Euler
line of 
)
if and only if 
lies on union of the circumcircle and the Neuberg
cubic (Morley and Morley 1931).
The Neuberg cubic passes through the circular points at infinity.
