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 Gibert's Catalogue of Triangle Cubics which is a compilation of extensive information about more than 1200 triangle cubics.

Definitions

Neuberg cubic of triangle Template:Math showing one of the defining properties of an arbitrary point Template:Mvar on the curve

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 Template:Mvar in the plane of the reference triangle Template:Math such that, if the reflections of Template:Mvar in the sidelines of triangle Template:Math are Template:Mvar, then the lines Template:Mvar 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 Template:Mvar such that if Template:Mvar are the circumcenters of triangles Template:Math, then the lines Template:Mvar are concurrent. Yet another way is to define it as the locus of Template:Mvar satisfying the following property known as the quadrangles involutifs (this was the way in which Neuberg introduced the curve):

$\left|\begin{array}{ccc}1& B{C}^2+A{P}^2& B{C}^2\times A{P}^2\\ 1& C{A}^2+B{P}^2& C{A}^2\times B{P}^2\\ 1& A{B}^2+C{P}^2& A{B}^2\times C{P}^2\end{array}\right|=0$

Equation

Let Template:Mvar be the side lengths of the reference triangle Template:Math. Then the equation of the Neuberg cubic of Template:Math in barycentric coordinates Template:Math is

$\sum _{\text{cyclic}}[a^2(b^2+c^2)-(b^2-c^2{)}^2-2a^4]x(c^2{y}^2-b^2{z}^2)=0$

Other terminology: 21-point curve, 37-point curve

Neuberg cubic (21-point cubic) of triangle Template:Math showing the 21-point special points on it

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 Template:Math, the 21 points are as listed below.

• The vertices Template:Mvar
• The reflections Template:Mvar of the vertices Template:Mvar in the opposite sidelines
• The orthocentre Template:Mvar
• The circumcenter Template:Mvar
• The three points Template:Mvar where Template:Mvar is the reflection of A in the line joining Template:Mvar and Template:Mvar where Template:Mvar is the intersection of the perpendicular bisector of Template:Mvar with Template:Mvar and Template:Mvar is the intersection of the perpendicular bisector of Template:Mvar with Template:Mvar; Template:Mvar and Template:Mvar are defined similarly
• The six vertices Template:Mvar of the equilateral triangles constructed on the sides of triangle Template:Math
• 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)

The attached figure shows the Neuberg cubic of triangle Template:Math 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. 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. Gibert's Catalogue has a special page dedicated to a listing of such special points which are also triangle centers.

Some properties of the Neuberg cubic

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

$\begin{array}{cc}& \cos B+i\sin B:\cos A-i\sin A:-1\\ & \cos B-i\sin B:\cos A+i\sin A:-1\end{array}$

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 Template:Mvar with respect to a triangle Template:Math is the point of concurrence of the reflections of the lines Template:Mvar about the angle bisectors of Template:Mvar respectively. The isogonal conjugate of Template:Mvar is sometimes denoted by Template:Mvar. The isogonal conjugate of Template:Mvar is Template:Mvar. A self-isogonal cubic is a triangle cubic that is invariant under isogonal conjugation. A pivotal isogonal cubic is a cubic in which points Template:Mvar lying on the cubic and their isogonal conjugates are collinear with a fixed point Template:Mvar 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 Template:Mvar be a point in the plane of triangle Template:Math. The perpendicular lines at Template:Mvar to Template:Mvar intersect Template:Mvar respectively at Template:Mvar and these points lie on a line Template:Mvar. Let the trilinear pole of Template:Mvar be Template:Math. An isopivotal cubic is a triangle cubic having the property that there is a fixed point Template:Mvar such that, for any point M on the cubic, the points Template:Math are collinear. The fixed point Template:Mvar is called the orthopivot of the cubic. The Neuberg cubic is an orthopivotal cubic with orthopivot at the triangle's circumcenter.

Additional reading

• Template:Cite journal
• Template:Cite journal
• Abdikadir Altintas, On Some Properties Of Neuberg Cubic
• Template:Cite web