Neuberg cubic

In Euclidean geometry, the Neuberg cubic is a special cubic plane curve associated with a reference triangle with several remarkable properties. 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. The curve appears as the first item, with identification number K001, in Bernard Gilbert's Catalogue of Triangle Cubics which is a compilation of extensive information about more than 1200 triangle cubics. The Neuberg cubic can be defined as a locus in many different ways. 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 P_a, P_b, P_c, then the lines AP_a, BP_b, CP_c are concurrent. However, it needs to be proved that the locus so defined is indeed a cubic curve. A second way is to define it as the locus of point P such that if O_a, O_b, O_c are the circumcenters of triangles △BPC, △CPA, △APB, then the lines AO_a, BO_b, O_c are concurrent. Yet another way is to define it as the locus of P satisfying the following property known as the quadrangles involutifs (this was the way in which Neuberg introduced the curve): Let a, b, c be the side lengths of the reference triangle △ABC. Then the equation of the Neuberg cubic of △ABC in barycentric coordinates x : y : z is In the older literature the Neuberg curve commonly referred to as the 21-point curve. The terminology refers to the property of the curve discovered by Neuberg himself that it passes through certain special 21 points associated with the reference triangle. Assuming that the reference triangle is △ABC, the 21 points are as listed below.