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.

Definitions


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 $X$ in the plane of the reference triangle $△ABC$ such that, if the reflections of $P$ in the sidelines of triangle $△ABC$ are $P$, then the lines $Pa, Pb, Pc$ 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 $APa, BPb, CPc$ such that if $P$ are the circumcenters of triangles $△ABC$, then the lines $Oa, Ob, Oc$ are concurrent. Yet another way is to define it as the locus of $AOa, BOb, Oc$ satisfying the following property known as the quadrangles involutifs (this was the way in which Neuberg introduced the curve):
 * $$\begin{vmatrix}

1 & BC^2+AP^2 & BC^2\times AP^2 \\ 1 & CA^2+BP^2 & CA^2\times BP^2\\ 1 & AB^2+CP^2& AB^2\times CP^2 \end{vmatrix} = 0$$

Equation
Let $P$ be the side lengths of the reference triangle $△BPC, △CPA, △APB$. Then the equation of the Neuberg cubic of $△ABC$ in barycentric coordinates $△ABC$ is
 * $$ \sum_{\text{cyclic}} [a^2(b^2+c^2)- (b^2-c^2)^2 -2a^4]x(c^2y^2 - b^2z^2)=0$$

Other terminology: 21-point curve, 37-point curve
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 $x : y : z$, the 21 points are as listed below. The attached figure shows the Neuberg cubic of triangle $△ABC$ with all the above mentioned 21 special points on it.
 * The vertices $a, b, c$
 * The reflections $A, B, C$ of the vertices $Aa, Bb, Cc$ in the opposite sidelines
 * The orthocentre $A, B, C$
 * The circumcenter $H$
 * The three points $O$ where $Da, Db, Dc$ is the reflection of A in the line joining $Da$ and $Qbc$ where $Qcb$ is the intersection of the perpendicular bisector of $Qbc$ with $AC$ and $AB$ is the intersection of the perpendicular bisector of $Qcb$ with $AB$; $AC$ and $Db$ are defined similarly
 * The six vertices $Dc$ of the equilateral triangles constructed on the sides of triangle $△ABC$
 * The two isogonic centers (the points X(13) and X(14) in the Encyclopedia of Triangle Centers)
 * The two isodynamic points (the points X(15) and X(16) in the Encyclopedia of Triangle Centers)

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. Because of this, the Neuberg cubic is also sometimes referred to as the 37-point cubic. Currently, a huge number of special points are known to lie on the Neuberg cubic. Gilbert's Catalogue has a special page dedicated to a listing of such special points which are also triangle centers.

Neuberg cubic as a circular cubic
The equation in trilinear coordinates of the line at infinity in the plane of the reference triangle is
 * $$ax+by+cz=0$$

There are two special points on this line called the circular points at infinity. Every circle in the plane of the triangle passes through these two points and every conic which passes through these points is a circle. The trilinear coordinates of these points are
 * $$\begin{align}

& \cos B + i\sin B : \cos A - i\sin A : -1 \\ & \cos B-i\sin B : \cos A+i\sin A: -1 \end{align}$$ where $$i=\sqrt{-1}$$. Any cubic curve which passes through the two circular points at infinity is called a circular cubic. The Neuberg cubic is a circular cubic.

Neuberg cubic as a pivotal isogonal cubic
The isogonal conjugate of a point $A', B', C', A", B", C"$ with respect to a triangle $△ABC$ is the point of concurrence of the reflections of the lines $P$ about the angle bisectors of $PA, PB, PC$ respectively. The isogonal conjugate of $A, B, C$ is sometimes denoted by $P$. The isogonal conjugate of $P*$ is $P*$. A self-isogonal cubic is a triangle cubic that is invariant under isogonal conjugation. A pivotal isogonal cubic is a cubic in which points $P$ lying on the cubic and their isogonal conjugates are collinear with a fixed point $P$ known as the pivot point of the cubic. The Neuberg cubic is a pivotal isogonal cubic having its pivot at the intersection of the Euler line with the line at infinity. In Kimberling's Encyclopedia of Triangle Centers, this point is denoted by X(30).

Neuberg cubic as a pivotol orthocubic
Let $Q$ be a point in the plane of triangle $△ABC$. The perpendicular lines at $P$ to $P$ intersect $AP, BP, CP$ respectively at $BC, CA, AB$ and these points lie on a line $Pa, Pb, Pc$. Let the trilinear pole of $LP$ be $△ABC$. An isopivotal cubic is a triangle cubic having the property that there is a fixed point $LP$ such that, for any point M on the cubic, the points $△ABC$ are collinear. The fixed point $P$ is called the orthopivot of the cubic. The Neuberg cubic is an orthopivotal cubic with orthopivot at the triangle's circumcenter.

Additional reading

 * Abdikadir Altintas, On Some Properties Of Neuberg Cubic
 * Abdikadir Altintas, On Some Properties Of Neuberg Cubic
 * Abdikadir Altintas, On Some Properties Of Neuberg Cubic