Neuberg cubic

It is named after Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926), a Luxembourger mathematician, who first introduced the curve in a paper published in 1884.

[1] One way is to define it as a locus of a point P in the plane of the reference triangle △ABC such that, if the reflections of P in the sidelines of triangle △ABC are Pa, Pb, Pc, then the lines APa, BPb, CPc are concurrent.

[3] The attached figure shows the Neuberg cubic of triangle △ABC with all the above mentioned 21 special points on it.

In a paper published in 1925, B. H. Brown reported his discovery of 16 additional special points on the Neuberg cubic making the total number of then known special points on the cubic 37.

Currently, a huge number of special points are known to lie on the Neuberg cubic.

[4] The equation in trilinear coordinates of the line at infinity in the plane of the reference triangle is There are two special points on this line called the circular points at infinity.

The perpendicular lines at P to AP, BP, CP intersect BC, CA, AB respectively at Pa, Pb, Pc and these points lie on a line LP.