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. 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.

Coordinates

Suppose that is a triangle with sidelengths Relative to, 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:
to convert from barycentric to trilinear, use
Many equations for cubics have the form
In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:
The cubics listed below can be defined in terms of the isogonal conjugate, denoted by, of a point not on a sideline of. A construction of follows. Let be the reflection of line about the internal angle bisector of angle, and define and analogously. Then the three reflected lines concur in. In trilinear coordinates, if then

First few triangle cubics in the catalogue

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

Identification number	Name	Equation in barycentric coordinates
	Neuberg cubic, 21-point cubic, 37-point cubic
	Thomson cubic, 17-point cubic
	McCay cubic, Griffiths cubic
	Darboux cubic
	[\|Napoleon cubic], Feuerbach cubic
	Orthocubic
	[\|Lucas cubic]
	Droussent cubic
	Lemoine cubic
	Simson cubic

Individual curves

Neuberg cubic

Trilinear equation:
Barycentric equation:
The Neuberg cubic is the locus of a point such that is on the line, where is the Euler infinity point. Also, this cubic is the locus of such that the triangle is perspective to, where is the reflection of in the lines 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 in the sidelines of, and the vertices of the six equilateral triangles erected on the sides of.
For a graphical representation and extensive list of properties of the Neuberg cubic, see .

Thomson cubic

Trilinear equation:
Barycentric equation:
The Thomson cubic is the locus of a point such that is on the line, where is the centroid.
The Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the vertices the excenters, the midpoints of sides and the midpoints of the altitudes of. For each point on the cubic but not on a sideline of, the isogonal conjugate of is also on the cubic.
For graphs and properties, see .

Darboux cubic

Trilinear equation:
Barycentric equation:
The Darboux cubic is the locus of a point such that is on the line, where is the de Longchamps point. Also, this cubic is the locus of such that the pedal triangle of is the cevian triangle of some point. Also, this cubic is the locus of a point such that the pedal triangle of and the anticevian triangle of 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 the excenters, and the antipodes of on the circumcircle. For each point on the cubic but not on a sideline of, the isogonal conjugate of is also on the cubic.
For graphics and properties, see .

Napoleon–Feuerbach cubic

Trilinear equation:
Barycentric equation:
The Napoleon–Feuerbach cubic is the locus of a point is on the line, where is the nine-point center,.
The Napoleon–Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the vertices the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of.
For a graphics and properties, see .

Lucas cubic

Trilinear equation:
Barycentric equation:
The Lucas cubic is the locus of a point such that the cevian triangle of 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 .

1st Brocard cubic

Trilinear equation:
Barycentric equation:
Let be the 1st Brocard triangle. For arbitrary point, let be the intersections of the lines with the sidelines respectively. The 1st Brocard cubic is the locus of for which the points 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 .

2nd Brocard cubic

Trilinear equation:
Barycentric equation:
The 2nd Brocard cubic is the locus of a point for which the pole of the line in the circumconic through and lies on the line of the circumcenter and the symmedian point. 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 .

1st equal areas cubic

Trilinear equation:
Barycentric equation:
The 1st equal areas cubic is the locus of a point such that area of the cevian triangle of equals the area of the cevian triangle of. Also, this cubic is the locus of for which is on the line, where is the Steiner point..
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 .

2nd equal areas cubic

Trilinear equation:
Barycentric equation:
For any point , let and The 2nd equal areas cubic is the locus of such that the area of the cevian triangle of equals the area of the cevian triangle of.
The 2nd equal areas cubic passes through the incenter, centroid, symmedian point, and points in Encyclopedia of Triangle Centers indexed as X, X, X, X, X, X, X, X, X, and others.
For a graphics and properties, see .

GeoGebra tool to draw triangle cubics

, 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

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''