Neuberg cubic

Last updated November 11, 2022

In mathematics, in triangle geometry, Neuberg cubic is a special cubic plane curve in the plane of the reference triangle having several remarkable properties. It is a triangle cubic in that it is associated with the reference triangle. 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]^[2] The curve appears as the first item, with identification number K001,^[1] in Bernard Gilbert's Catalogue of Triangle Cubics which is a compilation of extensive information about more than 1200 triangle cubics.

Definitions

Neuberg cubic of triangle ABC showing one of the defining properties of an arbitrary point X on the curve NeubergCurve.png — Neuberg cubic of triangle ABC showing one of the defining properties of an arbitrary point X on the curve

The Neuberg cubic can be defined as a locus in many different ways.^[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 P_a, P_b, P_c, then the lines AP_a, BP_b and 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 and APB, then the lines AO_a, BO_b and 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^[1] (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 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

\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

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.^[3]

The vertices A, B, C
The reflections A_a, B_b, C_c of the vertices A, B, C in the opposite sidelines
The orthocentre H
The circumcenter O
The three points D_a, D_b, D_c where D_a is the reflection of A in the line joining Q_bc and Q_cb where Q_bc is the intersection of the perpendicular bisector of AC with AB and Q_cb is the intersection of the perpendicular bisector of AB with AC; D_b and D_c are defined similarly
The six vertices A', B', C', A' ', B' ' and C' ' 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)

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.^[3] 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.^[4]

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

(\cos B+i\sin B:\cos A-i\sin A:-1),(\cos B-i\sin B:\cos A+i\sin A:-1)

where $i={\sqrt {-1}}$ .^[5] Any cubic curve which passes through the two circular points at infinity is called a circular cubic. The Neuberg cubic is a circular cubic.^[1]

Neuberg cubic as a pivotal isogonal cubic

The isogonal conjugate of a point P with respect to a triangle ABC is the point of concurrence of the reflections of the lines PA, PB, and PC about the angle bisectors of A, B, and C respectively. The isogonal conjugate of P 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 Q 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 P be a point in the plane of triangle ABC. The perpendicular lines at P to AP, BP, CP intersect BC, CA, AB respectively at P_a, P_b, P_c and these points lie on a line L_P. Let the trilinear pole of L_P be P^$\perp$. An isopivotal cubic is a triangle cubic having the property that there is a fixed point P such that, for any point M on the cubic, the points P, M, M^$\perp$ are collinear. The fixed point P is called the orthopivot of the cubic.^[6] The Neuberg cubic is an orthopivotal cubic with orthopivot at the triangle's circumcenter.^[1]

Additional reading

Zvonko Čerin (1998). "Locus properties of the Neuberg cubic". Journal of Geometry. 63 (1–2): 39–56. doi:10.1007/BF01221237. S2CID 116778499 . Retrieved 30 November 2021.
T. W. Moore and J. H. Neelley (May 1925). "The Circular Cubic on Twenty-One Points of a Triangle". The American Mathematical Monthly. 32 (5): 241–246. doi:10.1080/00029890.1925.11986451. JSTOR 2299192 . Retrieved 2 December 2021.
Abdikadir Altintas, On Some Properties Of Neuberg Cubic
Bernard Gilbert. "Neuberg Cubics" (PDF). Cubics in the Triangle Plane. Retrieved 2 December 2021.

Related Research Articles

In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.

<span class="mw-page-title-main">Altitude (triangle)</span> Perpendicular line segment from a triangles side to opposite vertex

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.

In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

<span class="mw-page-title-main">Euler line</span> Line constructed from a triangle

In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.

<span class="mw-page-title-main">Incenter</span> Center of the inscribed circle of a triangle

In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

In geometry, symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle, and reflecting the line over the corresponding angle bisector. The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.

<span class="mw-page-title-main">Cubic plane curve</span> Type of a mathematical curve

In mathematics, a cubic plane curve is a plane algebraic curve $C$ defined by a cubic equation

In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.

In geometry, the isogonal conjugate of a point $P$ with respect to a triangle $ABC$ is constructed by reflecting the lines $PA$ , $PB$ , and $PC$ about the angle bisectors of $A$ , $B$ , and $C$ respectively. These three reflected lines concur at the isogonal conjugate of $P$ . This is a direct result of the trigonometric form of Ceva's theorem.

In geometry, the trilinear coordinates $x : y : z$ of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio $x : y$ is the ratio of the perpendicular distances from the point to the sides opposite vertices $A$ and $B$ respectively; the ratio $y : z$ is the ratio of the perpendicular distances from the point to the sidelines opposite vertices $B$ and $C$ respectively; and likewise for $z : x$ and vertices $C$ and $A$ .

In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices. The nine-point center is listed as point X(5) in Clark Kimberling's Encyclopedia of Triangle Centers.

In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle. Triangles that are similar to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are triangle centers, and unlike other triangle centers the isodynamic points are also invariant under Möbius transformations. A triangle that is itself equilateral has a unique isodynamic point, at its centroid(as well as its orthocenter, its incenter, and its circumcenter, which are concurrent); every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by Joseph Neuberg (1885).

In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician.

In geometry, a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.

In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.

In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994.

In geometry, the Feuerbach hyperbola is a rectangular hyperbola passing through important triangle centers such as the Orthocenter, Gergonne point, Nagel point and Shiffler point. The center of the hyperbola is the Feuerbach point, the point of tangency of the incircle and the nine-point circle.

In triangle geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steiner ellipse which is an ellipse passing through the vertices and having its centre at the centroid of the reference triangle, the Kiepert hyperbola which is a conic passing through the vertices, the centroid and the orthocentre of the reference triangle and the Artzt parabolas which are parabolas touching two sidelines of the reference triangle at vertices of the triangle. The terminology of triangle conic is widely used in the literature without a formal definition,that is, without precisely formulating the relations a conic should have with the reference triangle so as to qualify it to be called a triangle conic (see,). WolframMathWorld has a page titled "Triangle conics" which gives a list of 42 items without giving a definition of triangle conic. However, Paris Pamfilos in his extensive collection of topics in geometry and topics in other fields related to geometry defines a triangle conic as a "conic circumscribing a triangle ABC or inscribed in a triangle ". The terminology triangle circle is used to denote a circle associated with the reference triangle is some way.

In mathematics, in triangle geometry, McCay cubic is a cubic plane curve in the plane of the reference triangle and associated with it, and having several remarkable properties. It is the third cubic curve in Bernard Gilbert's Catalogue of Triangle Cubics and it is assigned the identification number K003.

<span class="mw-page-title-main">Modern triangle geometry</span>

In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the time of Euclid. In fact, Euclid's Elements contains description of the four special points – centroid, incenter, circumcenter and orthocenter - associated with a triangle. Even though Pascal and Ceva in the seventeenth century, Euler in the eighteenth century and Feuerbach in the nineteenth century and many other mathematicians had made important discoveries regarding the properties of the triangle, it was the publication in 1873 of a paper by Emile Lemoine (1840–1912) with the title "On a remarkable point of the triangle" that was considered to have, according to Nathan Altschiller-Court, "laid the foundations...of the modern geometry of the triangle as a whole." The American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these [geometers] more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the properties of the triangle during the last quarter of the nineteenth century and the early years of the twentieth century. A hundred-page article on triangle geometry in Klein's Encyclopedia of Mathematical Sciences published in 1914 bears witness to this upsurge of interest in triangle geometry.

References

1 2 3 4 5 6 "K001 Neuberg cubic". Cubics in the Triangle Plane. Bernard Gilbert. Retrieved 29 November 2021.
↑ "Mémoire sur le tétraèdre". Mémoires de l'Académie de Belgique: 1–70. 1884. Retrieved 29 November 2021.
1 2 B H Brown (March 1925). "The 21-point Cubic". The American Mathematical Monthly. 35 (3): 110–115. doi:10.1080/00029890.1925.11986425.
↑ Bernard Gilbert. "Table 19: points on the Neuberg cubic". Cubics in the Triangle Plane. Bernard Gilbert. Retrieved 1 December 2021.
↑ Whitworth William Allen (1866). Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions. Deighton Bell And Company. p. 127. Retrieved 8 December 2021.
↑ Bernard Gibert (2003). "Orthocorrespondence and Orthopivotal Cubics". Forum Geometricorum. 3: 1–27. Retrieved 9 December 2021.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[K001-1] 1 2 3 4 5 6 "K001 Neuberg cubic". Cubics in the Triangle Plane. Bernard Gilbert. Retrieved 29 November 2021.

[2] "Mémoire sur le tétraèdre". Mémoires de l'Académie de Belgique: 1–70. 1884. Retrieved 29 November 2021.

[Brown-3] 1 2 B H Brown (March 1925). "The 21-point Cubic". The American Mathematical Monthly. 35 (3): 110–115. doi:10.1080/00029890.1925.11986425.

[4] Bernard Gilbert. "Table 19: points on the Neuberg cubic". Cubics in the Triangle Plane. Bernard Gilbert. Retrieved 1 December 2021.

[5] Whitworth William Allen (1866). Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions. Deighton Bell And Company. p. 127. Retrieved 8 December 2021.

[6] Bernard Gibert (2003). "Orthocorrespondence and Orthopivotal Cubics". Forum Geometricorum. 3: 1–27. Retrieved 9 December 2021.

[1]

[2]

[3]

[4]

[5]

[6]