Neuberg Cubic


The Neuberg cubic Z(X_(30)) of a triangle DeltaABC is the locus of all points P whose reflections in the sidelines BC, CA, and ABform a triangle perspective to DeltaABC. It is a self-isogonal cubic with pivot point at the Euler infinity point X_(30), so it has parameter x=cosA-2cosBcosC and trilinear equation


(Cundy and Parry 1995).

It passes through Kimberling centers X_i for i=1 (incenter I), 3 (circumcenter O), 4 (orthocenter H), 13 (first Fermat point X), 14 (second Fermat point X^'), 15 (first isodynamic point S), 16 (second isodynamic point S^'), 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 J_A, J_B, and J_C of the reference triangle and the circular points at infinity.

Let P be a triangulation point and let O_a be the circumcenter of DeltaPBC, O_b the circumcenter of DeltaAPC and O_c the circumcenter of DeltaABP. Then lines AO_a, BO_b and CO_c are concurrent (at a point on the line joining the circumcenter of ABC and P) if and only if P lies on the union of the circumcircle and the Neuberg cubic (Neuberg 1884).

Let l_a be the Euler line of DeltaPBC, l_b the Euler line of DeltaAPC and l_c the Euler line of DeltaABP. The lines l_a, l_b and l_c are concurrent (at a point on the Euler line of DeltaABC) if and only if P lies on union of the circumcircle and the Neuberg cubic (Morley and Morley 1931).

The Neuberg cubic passes through the circular points at infinity.

See also

Self-Isogonal Cubic, Triangle Cubic

Portions of this entry contributed by Floor van Lamoen

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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.", A. P.; van Lamoen, F. M.; Wolk, B.; and Yiu, P. "Concurrency of Four Euler Lines." Forum Geom. 1, 59-68, 2001., 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.

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Neuberg Cubic

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Neuberg Cubic." From MathWorld--A Wolfram Web Resource.

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