Modern triangle geometry

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.

Here is a definition of triangle geometry from 1887: "Being given a point M in the plane of the triangle, we can always find, in an infinity of manners, a second point M' that corresponds to the first one according to an imagined geometrical law; these two points have between them geometrical relations whose simplicity depends on the more or less the lucky choice of the law which unites them and each geometrical law gives rise to a method of transformation a mode of conjugation which it remains to study."

(See the conference paper titled "Teaching new geometrical methods with an ancient figure in the nineteenth and twentieth centuries: the new triangle geometry in textbooks in Europe and USA (1888–1952)" by Pauline Romera-Lebret presented in 2009.

[6]) However, this escalation of interest soon collapsed and triangle geometry was completely neglected until the closing years of the twentieth century.

In his "Development of Mathematics", Eric Temple Bell offers his judgement on the status of modern triangle geometry in 1940 thus: "The geometers of the 20th Century have long since piously removed all these treasures to the museum of geometry where the dust of history quickly dimmed their luster."

(The Development of Mathematics, p. 323)[5] Philip Davis has suggested several reasons for the decline of interest in triangle geometry.

[5] These include: A further revival of interest was witnessed with the advent of the modern electronic computer.

The triangle geometry has again become an active area of research pursued by a group of dedicated geometers.

For a given triangle ABC with centroid G, the symmedian through the vertex is the reflection of the line AG in the bisector of the angle A.

In modern triangle geometry, there is a large body of literature dealing with properties of orthopoles.

[20][21] Let of circles be described on the sides BC, CA, AB of triangle ABC whose external segments contain the two triads of angles C, A, B and B, C, A respectively.

This concept introduced by Clark Kimberling in 1994 unified in one notion the very many special and remarkable points associated with a triangle.

For example, the 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 Pa, Pb, Pc, then the lines APa, BPb and CPc are concurrent is a cubic curve named Neuberg cubic.

[28] A survey article published in 2015 gives an account of some of the important new results discovered by the computer programme "Discoverer".

Sava Grozdev, Hiroshi Okumura, Deko Dekov are maintaining a web portal dedicated to computer discovered encyclopedia of Euclidean geometry.