Catalogue of Triangle Cubics explained

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.^[1] The resource is maintained by Bernard Gilbert. 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.^[2]

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 _{cyclic} [a ^{2(b} ^{2+c} ^{2)-} (b ^{2-c} ^{2)} ^{2} -2 a ^{4]x(c} ^{2y} ^{2} - b ^{2z} ^{2)=0}
K002	Thomson cubic, 17-point cubic	\sum _{cyclic} x(c ^{2y} ^{2} - b ^{2z} ^{2)=0}
K003	McCay cubic, Griffiths cubic	\sum _{cyclic} a ^{2(b} ^{2+c} ^{2-a} ^{2)x(c} ^{2y} ^{2} - b ^{2z} ^{2)=0}
K004		\sum _{cyclic} [ 2 a ^{2(b} ^{2+c} ^{2)-} (b ^{2-c} ^{2)} ^{2} -3 a ^{4]x(c} ^{2y} ^{2} - b ^{2z} ^{2)=0}
K005	Napoleon cubic, Feuerbach cubic	\sum _{cyclic} [a ^{2(b} ^{2+c} ^{2)-} (b ^{2-c} ^{2)} ^{2]x(c} ^{2y} ^{2} - b ^{2z} ^{2)=0}
K006	Orthocubic	\sum _{cyclic} (c ^{2+a} ^{2-b} ^{2)(a} ^{2+b} ^{2-c} ^{2)x(c} ^{2y} ^{2} - b ^{2z} ^{2)=0}
K007	Lucas cubic	\sum _{cyclic} (b ^{2+c} ^{2-a} ^{2)x(y} ^{2-z} ^{2)=0}
K008	Droussent cubic	\sum _{cyclic} (b ^{4+c} ^{4-a} ^{4-b} ^{2c} ^{2)x(y} ^{2-z} ^{2)=0}
K009	Lemoine cubic	\begin{align}& 2 (a ^{2-b} ^{2)(b} ^{2-c} ^{2)(c} ^{2-a} ^{2 )xyz\\&\sum} ^{cyclic} a ^{4(b} ^{2+c} ^{2-a} ^{2)yz(y-z)=0\end{align}}
K010	Simson cubic	\sum _{cyclic} a ^{2 (y + z)/(y - z)} =0

GeoGebra tool to draw triangle cubics

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.^[3] The command

• Cubic(, , , 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

Notes and References

1. Web site: Bernard Gilbert . Catalogue of Triangle Cubics . Cubics in the Triangle Plane . Bernard Gilbert . 27 November 2021.
2. Web site: K1200: a crunodal KHO-cubic . Cubics in the Trangle Plane . Bernard Gilbert . 27 November 2021.
3. Web site: Cubic Command . GeoGebra . GeoGebra . 27 November 2021.