Catalogue of Triangle Cubics

The Catalogue of Triangle Cubics is an online resource containing detailed information about more than 1200 cubic curves in the plane of a reference triangle. The resource is maintained by Bernard Gibert. Each cubic in the resource is assigned a unique identification number of the form "Knnn" where "nnn" denotes three digits. The identification number of the first entry in the catalogue is "K001" which is the Neuberg cubic of the reference triangle Template:Mvar. The catalogue provides, among other things, the following information about each of the cubics listed:

• Barycentric equation of the curve
• A list of triangle centers which lie on the curve
• Special points on the curve which are not triangle centers
• Geometric properties of the curve
• Locus properties of the curve
• Other special properties of the curve
• Other curves related to the cubic curve
• Plenty of neat and tidy figures illustrating the various properties
• References to literature on the curve

The equations of some of the cubics listed in the Catalogue are so incredibly complicated that the maintainer of the website has refrained from putting up the equation in the webpage of the cubic; instead, a link to a file giving the equation in an unformatted text form is provided. For example, the equation of the cubic K1200 is given as a text file.

Suppose that Template:Math is a triangle with sidelengths $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Relative to Template:Math, many named cubics pass through well-known points. Examples shown below use two kinds of homogeneous coordinates: trilinear and barycentric.

To convert from trilinear to barycentric in a cubic equation, substitute as follows:

$x\to bcx,y\to cay,z\to abz;$

to convert from barycentric to trilinear, use

$x\to ax,y\to by,z\to cz.$

Many equations for cubics have the form

$f(a,b,c,x,y,z)+f(b,c,a,y,z,x)+f(c,a,b,z,x,y)=0.$

In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:

$\sum _{\text{cyclic}}f(x,y,z,a,b,c)=0$.

The cubics listed below can be defined in terms of the isogonal conjugate, denoted by Template:Mvar, of a point Template:Mvar not on a sideline of Template:Math. A construction of Template:Mvar follows. Let Template:Mvar be the reflection of line Template:Mvar about the internal angle bisector of angle Template:Mvar, and define Template:Mvar and Template:Mvar analogously. Then the three reflected lines concur in Template:Mvar. In trilinear coordinates, if $X=x:y:z,$ then $X^{\ast }=\frac{1}{x}:\frac{1}{y}:\frac{1}{z}.$

First few triangle cubics in the catalogue

The following are the first ten cubics given in the Catalogue.

Identification number	Name(s)	Equation in barycentric coordinates
K001	Neuberg cubic, 21-point cubic, 37-point cubic	$\sum _{\text{cyclic}}[a^2(b^2+c^2)-(b^2-c^2{)}^2-2a^4]x(c^2{y}^2-b^2{z}^2)=0$
K002	Thomson cubic, 17-point cubic	$\sum _{\text{cyclic}}x(c^2{y}^2-b^2{z}^2)=0$
K003	McCay cubic, Griffiths cubic	$\sum _{\text{cyclic}}{a}^2(b^2+c^2-a^2)x(c^2{y}^2-b^2{z}^2)=0$
K004	Darboux cubic	$\sum _{\text{cyclic}}[2a^2(b^2+c^2)-(b^2-c^2{)}^2-3a^4]x(c^2{y}^2-b^2{z}^2)=0$
K005	Napoleon cubic, Feuerbach cubic	$\sum _{\text{cyclic}}[a^2(b^2+c^2)-(b^2-c^2{)}^2]x(c^2{y}^2-b^2{z}^2)=0$
K006	Orthocubic	$\sum _{\text{cyclic}}(c^2+a^2-b^2)(a^2+b^2-c^2)x(c^2{y}^2-b^2{z}^2)=0$
K007	Lucas cubic	$\sum _{\text{cyclic}}(b^2+c^2-a^2)x(y^2-z^2)=0$
K008	Droussent cubic	$\sum _{\text{cyclic}}(b^4+c^4-a^4-b^2{c}^2)x(y^2-z^2)=0$
K009	Lemoine cubic	$\begin{array}{cc}& 2(a^2-b^2)(b^2-c^2)(c^2-a^2)xyz\\ & \sum _{\text{cyclic}}{a}^4(b^2+c^2-a^2)yz(y-z)=0\end{array}$
K010	Simson cubic	$\sum _{\text{cyclic}}{a}^2\frac{y+z}{y-z}=0$

First six cubics in the Catalogue of Triangle Cubics

Individual curves

Neuberg cubic

Neuberg cubic of triangle Template:Math: The locus of Template:Mvar such that, if Template:Mvar are the reflections of Template:Mvar in the sidelines Template:Mvar, then the lines Template:Mvar are concurrent.

Trilinear equation: $\sum _{\text{cyclic}}(\cos A-2\cos B\cos C)x(y^2-z^2)=0$

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

The Neuberg cubic (named after Joseph Jean Baptiste Neuberg) is the locus of a point Template:Mvar such that Template:Mvar is on the line Template:Mvar, where Template:Mvar is the Euler infinity point (Template:Math in the Encyclopedia of Triangle Centers). Also, this cubic is the locus of Template:Mvar such that the triangle Template:Math is perspective to Template:Math, where Template:Math is the reflection of Template:Mvar in the lines Template:Mvar respectively

The Neuberg cubic passes through the following points: incenter, circumcenter, orthocenter, both Fermat points, both isodynamic points, the Euler infinity point, other triangle centers, the excenters, the reflections of Template:Mvar in the sidelines of Template:Math, and the vertices of the six equilateral triangles erected on the sides of Template:Math.

For a graphical representation and extensive list of properties of the Neuberg cubic, see K001 at Berhard Gibert's Cubics in the Triangle Plane.

Thomson cubic

Example of Thomson cubic (black curve). Template:Mvar is on the cubic, such that the isogonal conjugate of Template:Math is on the line Template:Math.

Trilinear equation: $\sum _{\text{cyclic}}bcx(y^2-z^2)=0$

Barycentric equation: $\sum _{\text{cyclic}}x(c^2{y}^2-b^2{z}^2)=0$

The Thomson cubic is the locus of a point Template:Mvar such that Template:Mvar is on the line Template:Mvar, where Template:Mvar is the centroid.

The Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the vertices Template:Mvar the excenters, the midpoints of sides Template:Mvar and the midpoints of the altitudes of Template:Math. For each point Template:Mvar on the cubic but not on a sideline of Template:Math, the isogonal conjugate of Template:Mvar is also on the cubic.

For graphs and properties, see K002 at Cubics in the Triangle Plane.

Darboux cubic

Darboux cubic of triangle Template:Math: The locus of Template:Mvar such that if Template:Mvar are the feet of perpendiculars from Template:Mvar to the sidelines Template:Mvar then the lines Template:Mvar are concurrent.

Trilinear equation:$\sum _{\text{cyclic}}(\cos A-\cos B\cos C)x(y^2-z^2)=0$

Barycentric equation: $\sum _{\text{cyclic}}(2a^2(b^2+c^2)+(b^2-c^2{)}^2-3a^4)x(c^2{y}^2-b^2{z}^2)=0$

The Darboux cubic is the locus of a point Template:Mvar such that Template:Mvar is on the line Template:Mvar, where Template:Mvar is the de Longchamps point. Also, this cubic is the locus of Template:Mvar such that the pedal triangle of Template:Mvar is the cevian triangle of some point (which lies on the Lucas cubic). Also, this cubic is the locus of a point Template:Mvar such that the pedal triangle of Template:Mvar and the anticevian triangle of Template:Mvar are perspective; the perspector lies on the Thomson cubic.

The Darboux cubic passes through the incenter, circumcenter, orthocenter, de Longchamps point, other triangle centers, the vertices Template:Mvar the excenters, and the antipodes of Template:Mvar on the circumcircle. For each point Template:Mvar on the cubic but not on a sideline of Template:Math, the isogonal conjugate of Template:Mvar is also on the cubic.

For graphics and properties, see K004 at Cubics in the Triangle Plane.

Napoleon–Feuerbach cubic

Trilinear equation: $\sum _{\text{cyclic}}\cos (B-C)x(y^2-z^2)=0$

Barycentric equation: $\sum _{\text{cyclic}}(a^2(b^2+c^2)+(b^2-c^2{)}^2)x(c^2{y}^2-b^2{z}^2)=0$

The Napoleon–Feuerbach cubic is the locus of a point Template:Mvar is on the line Template:Mvar, where Template:Mvar is the nine-point center, (Template:Math in the Encyclopedia of Triangle Centers).

The Napoleon–Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the vertices Template:Mvar the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of Template:Math.

For a graphics and properties, see K005 at Cubics in the Triangle Plane.

Lucas cubic

Lucas Cubic of triangle Template:Math: The locus of a point Template:Mvar such that the cevian triangle of Template:Mvar is the pedal triangle of some point Template:Mvar; the point Template:Mvar lies on the Darboux cubic.

Trilinear equation: $\sum _{\text{cyclic}}\cos (A)x(b^2{y}^2-c^2{z}^2)=0$

Barycentric equation: $\sum _{\text{cyclic}}(b^2+c^2-a^2)x(y^2-z^2)=0$

The Lucas cubic is the locus of a point Template:Mvar such that the cevian triangle of Template:Mvar is the pedal triangle of some point; the point lies on the Darboux cubic.

The Lucas cubic passes through the centroid, orthocenter, Gergonne point, Nagel point, de Longchamps point, other triangle centers, the vertices of the anticomplementary triangle, and the foci of the Steiner circumellipse.

For graphics and properties, see K007 at Cubics in the Triangle Plane.

1st Brocard cubic

First Brocard Cubic: It is the locus of Template:Mvar such the intersections of Template:Mvar with the sidelines Template:Mvar where Template:Math is the first Brocard triangle of triangle Template:Math, are collinear. In the figure Template:Mvar and Template:Mvar are the first and second Brocard points.

Trilinear equation:$\sum _{\text{cyclic}}bc(a^4-b^2{c}^2)x(y^2+z^2)=0$

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

Let Template:Math be the 1st Brocard triangle. For arbitrary point Template:Mvar, let Template:Mvar be the intersections of the lines Template:Mvar with the sidelines Template:Mvar respectively. The 1st Brocard cubic is the locus of Template:Mvar for which the points Template:Mvar are collinear.

The 1st Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.

For graphics and properties, see K017 at Cubics in the Triangle Plane.

2nd Brocard cubic

Trilinear equation: $\sum _{\text{cyclic}}bc(b^2-c^2)x(y^2+z^2)=0$

Barycentric equation: $\sum _{\text{cyclic}}(b^2-c^2)x(c^2{y}^2+b^2{z}^2)=0$

The 2nd Brocard cubic is the locus of a point Template:Mvar for which the pole of the line Template:Mvar in the circumconic through Template:Mvar and Template:Mvar lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis). The cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers, and the vertices of the 2nd and 4th Brocard triangles.

For a graphics and properties, see K018 at Cubics in the Triangle Plane.

1st equal areas cubic

First equal area cubic of triangle Template:Math: The locus of a point Template:Mvar such that area of the cevian triangle of Template:Mvar equals the area of the cevian triangle of Template:Mvar.

Trilinear equation: $\sum _{\text{cyclic}}a(b^2-c^2)x(y^2-z^2)=0$

Barycentric equation: $\sum _{\text{cyclic}}{a}^2(b^2-c^2)x(c^2{y}^2-b^2{z}^2)=0$

The 1st equal areas cubic is the locus of a point Template:Mvar such that area of the cevian triangle of Template:Mvar equals the area of the cevian triangle of Template:Mvar. Also, this cubic is the locus of Template:Mvar for which Template:Mvar is on the line Template:Mvar, where Template:Mvar is the Steiner point. (Template:Math in the Encyclopedia of Triangle Centers).

The 1st equal areas cubic passes through the incenter, Steiner point, other triangle centers, the 1st and 2nd Brocard points, and the excenters.

For a graphics and properties, see K021 at Cubics in the Triangle Plane.

2nd equal areas cubic

Trilinear equation: $(bz+cx)(cx+ay)(ay+bz)=(bx+cy)(cy+az)(az+bx)$

Barycentric equation:$\sum _{\text{cyclic}}a(a^2-bc)x(c^3{y}^2-b^3{z}^2)=0$

For any point $X=x:y:z$ (trilinears), let $X_Y=y:z:x$ and $X_Z=z:x:y.$ The 2nd equal areas cubic is the locus of Template:Mvar such that the area of the cevian triangle of Template:Mvar equals the area of the cevian triangle of Template:Mvar.

The 2nd equal areas cubic passes through the incenter, centroid, symmedian point, and points in Encyclopedia of Triangle Centers indexed as X(31), X(105), X(238), X(292), X(365), X(672), X(1453), X(1931), X(2053), and others.

For a graphics and properties, see K155 at Cubics in the Triangle Plane.

GeoGebra tool to draw triangle cubics

Tucker cubic (cubic K011 in the Catalogue) of triangle ABC drawn using the GeoGebra command Cubic(A,B,C,11).

GeoGebra, the software package for interactive geometry, algebra, statistics and calculus application has a built-in tool for drawing the cubics listed in the Catalogue. The command

• Cubic( <Point>, <Point>, <Point>, n)

prints the n-th cubic in the Catalogue for the triangle whose vertices are the three points listed. For example, to print the Thomson cubic of the triangle whose vertices are A, B, C the following command may be issued:

• Cubic(A, B, C, 2)

See also

• Modern triangle geometry