Question about the encyclopedia of triangle centers

The Encyclopedia of Triangle Centers has developed a neat approximate way to check whether an entry about a particular point already exists. It involves checking the first coordinate of the normalized trilinears of the "new" point against a list of all points.

All entries in the ETC however also list interesting properties of points, e.g. "X(44004) lies on these lines: {2, 35311}, {20, 10620}, {3448, 9033}, {5965, 43768}, {15319, 15801}"

My question is where do people get the data about tens of thousands of points to determine this, presumable in an automated way? I cannot imagine that this comes out of random checks, and also it is difficult to imagine that hundreds of contributors have typed all known points' barycentric coordinates in their own code to facilitate automatic checks.

Solution 1:

The founders of "Encyclopedia of Triangle Centers" have mainly conducted an intelligent compilation and structuration of thousands of results found, considering the most important ones, a century ago...

This does not mean that, in particular with computers' help, a big number of new centers and/or properties haven't been discovered in the meanwhile.

I am going to explain it with a historical perspective.

In mid-18th century, only 5 remarkable triangle centers had been identified, the first four having 2 thousand years of "existence" (incenter, centroid, circumcenter, orthocenter), a fifth one having been added by Euler, the nine points circle center. In the ETC, these points are listed as the first ones (X(1)...X(5)).

In the first three quarters of the 19th century, some others were gradually obtained : Gergonne point (X(7)) Feuerbach point (X(11)), around 1820, Lemoine (or symmedian) point (X(6)) and Brocard points later, around 1880, etc. Please note that the ranking in ETC doesn't follow in the 19th century the "date of birth" of the points... It looks based on an estimated relative importance of these centers.

Then, a triangle-mania arose in the last quarter of the 19th century, with a first peak around 1890-1910, a second even more important peak between 1920 and 1950 with an incidence on secondary school syllabus, followed by a significant decrease. At that time, thousands of more or less amateur mathematicians have been hunting properties of triangles (some of them rather remote, not to say sterile), and in particular have found this myriad of centers with, for many of them, barycentric or trilinear coordinates, alignment properties as you mention one, concyclicities, etc.

The "new maths" wave around 1960-1970 with a largely exaggered ban of figure geometry, left few people working on triangle geometry. It must be said that the "game" had become scarcer... most triangle properties had been found... with some exception, for example "Conway circle". A regain of interest came in the 1990s, in particular due to possibilities offered by the computer.

Remark: Historicaly, remarkable points were defined using "pure geometry" also known as the "synthetic method". It is only gradually, through the slow adoption of barycentric (or areal) coordinates (introduced by Möbius in 1827) and/or trilinear coordinates (born some decades later in particular with their use in physics ; a first reference being by Gibbs in 1873) that algebraic approaches have proven their efficiency. Please note that classical cartesian coordinates are unable to render the ternary symmetry existing between the vertices/sides/angles of a triangle. One must not forget as well that the distinction made between metric, affine, projective invariance of the different points/lines, etc... wasn't familiar to the mind of these "amateur geometers" (although some of them were very smart !) at the eve of 20th century.

I end this part by showing (Fig. 1) a partial statistics of use of the words incenter and orthocenter (partial because the corpus is limited to books in the "Google books corpus" written in English). Many biases are to be taken into account, in particular the fact that the share of mathematical books has been shrinking during that time.

Fig. 1: Statistics of relative use of the word "incenter" and "orthocenter" from 1860 till now, obtained using Google Books Ngram Viewer. One can note the "new math" impact in the 1970s followed by a considerable renewal of interest at the end of 20th century (due in part to ETC site !). Please note that the keyword "triangle" displays similar trends though some of its uses are outside mathematics. Another remark: The names "orthocenter" and "incenter" where created in the 1860s.

Edit:

1. An interesting document about certification of ETC site using Coq.

2. For recent works (since year 2000) on triangle geometry in particular, see the interesting "Forum Geometricum" journal.

3. About the new trend at the end of 19th century, see the PhD thesis of Pauline Romera-Lebret "La nouvelle géométrie du triangle : passage d'une mathématique d'amateurs à une mathématique d'enseignants (1873-1929)" (Univ. of Nantes, France, 2009).

4. Another thesis (by E. Cook, Univ. of Florida, 1937) compiles the works of a certain J. Mackey. It gives an idea of the extreme meticulosity of some of the "founding fathers" (here, a little more than an amateur !) of what I have called triangle-mania.

5. Another interesting article by Ph.J. Davis, "The rise, fall, and possible transfiguration of triangle geometry: A mini-history" Amer. Math. Monthly , 102 (1995) pp. 204–214

6. About Conway's "elementary" accomplishments, see this answer https://mathoverflow.net/questions/357197/conways-lesser-known-results/357406#357406) and the surrounding ones.

7. Don't miss Douillet's glossary dealing with barycentric coordinates of remarkable "centers".

8. About the "new maths" episode, see the controversial paper of a famous geometer, Dan Pedoe here.