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# Neuberg Cubic

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.

Self-Isogonal Cubic, Triangle Cubic

Portions of this entry contributed by Floor van Lamoen

## References

Čerin, Z. "Locus Properties of the Neuberg Cubic." J. Geom. 73, 39-56, 1998.Čerin, Z. "The Neuberg Cubic in Locus Problems." Math. Pannonica 11, 109-124, 2000.Cundy, H. M. and Parry, C. F. "Some Cubic Curves Associated with a Triangle." J. Geom. 53, 41-66, 1995.Gibert, B. "Neuberg Cubic." http://perso.wanadoo.fr/bernard.gibert/Exemples/k001.html.Hatzipolakis, A. P.; van Lamoen, F. M.; Wolk, B.; and Yiu, P. "Concurrency of Four Euler Lines." Forum Geom. 1, 59-68, 2001. http://forumgeom.fau.edu/FG2001volume1/FG200109index.html.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Moore, T. W. and Neelley, J. H. "The Circular Cubic on Twenty-One Points of a Triangle." Amer. Math. Monthly 32, 241-246, 1925.Morley, F. "Note on Neuberg Cubic Curve." Amer. Math. Monthly 32, 407-411, 1925.Morley, F. and Morley, F. V. Inversive Geometry. Oxford, England: Oxford University Press, pp. 199-200, 1931.Neuberg, J. "Mémoire sur le tétraèdre." Bruxelles, Belgium: F. Hayez, 1884.

Neuberg Cubic

## Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Neuberg Cubic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NeubergCubic.html