Modern triangle geometry

In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a <a href="/facts/Triangle/cRXUwsMn">triangle</a> discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the time of <a href="/facts/Euclid/dHpVyL4R">Euclid</a>. In fact, Euclid's <a href="/facts/Euclid%2527s_Elements/nPKvCIxs">Elements</a> contains description of the four special points – <a href="/facts/Centroid/H6dDbg0h">centroid</a>, <a href="/facts/Incenter/e933kPfL">incenter</a>, <a href="/facts/Circumcenter/RafYT4am">circumcenter</a> and <a href="/facts/Orthocenter/z9H8fFMz">orthocenter</a> - associated with a triangle. Even though <a href="/facts/Blaise_Pascal/Huy5LenL">Pascal</a> and <a href="/facts/Giovanni_Ceva/bgkWb3RI">Ceva</a> in the seventeenth century, <a href="/facts/Leonhard_Euler/z2fsbjgj">Euler</a> in the eighteenth century and <a href="/facts/Ludwig_Feuerbach/CkhGw2Kl">Feuerbach</a> 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 <a href="/facts/Emile_Lemoine/YEpjsfFH">Emile Lemoine</a> (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." The <a href="/facts/American_Mathematical_Monthly/Te6CHvkg">American Mathematical Monthly</a>, in which much of Lemoine's work is published, declared that "To none of these [geometers] more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the properties of the triangle during the last quarter of the nineteenth century and the early years of the twentieth century. A hundred-page article on triangle geometry in <a href="/facts/Klein%2527s_Encyclopedia_of_Mathematical_Sciences/EvgW8aCB">Klein's Encyclopedia of Mathematical Sciences</a> published in 1914 bears witness to this upsurge of interest in triangle geometry.
In the early days, the expression "new triangle geometry" referred to only the set of interesting objects associated with a triangle like the <a href="/facts/Lemoine_point/2MWFKJSg">Lemoine point</a>, Lemoine circle, <a href="/facts/Brocard_circle/I0bFT6En">Brocard circle</a> and the <a href="/facts/Lemoine_line/YvJ8Y7Sx">Lemoine line</a>. Later the theory of correspondences which was an offshoot of the theory of geometric transformations was developed to give coherence to the various isolated results. With its development, the expression "new triangle geometry" indicated not only the many remarkable objects associated with a triangle but also the methods used to study and classify these objects. 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.)
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", <a href="/facts/Eric_Temple_Bell/00WMLxa1">Eric Temple Bell</a> 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) Philip Davis has suggested several reasons for the decline of interest in triangle geometry. These include:

<ul><li>The feeling that the subject is elementary and of low professional status.</li>
<li>The exhaustion of its methodologic possibilities.</li>
<li>The visual complexity of the so-called deeper results of the subject.</li>
<li>The downgrading of the visual in favor of the algebraic.</li>
<li>A dearth of connections to other fields.</li>
<li>Competition with other topics with a strong visual content like <a href="/facts/Tessellation/7BWDywXC">tessellations</a>, <a href="/facts/Fractal/sXj38JeN">fractals</a>, <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, etc.</li></ul>
A further revival of interest was witnessed with the advent of the modern electronic <a href="/facts/Computer/SQe1aK05">computer</a>. The triangle geometry has again become an active area of research pursued by a group of dedicated geometers. As epitomizing this revival, one can point out the formulation of the concept of a "<a href="/facts/Triangle_centre/FKfm090E">triangle centre</a>" and the compilation by Clark Kimberling of an <a href="/facts/Encyclopedia_of_Triangle_Centers/ClS9kJor">encyclopedia</a> of triangle centers containing a listing of nearly 50,000 triangle centers and their properties and also the compilation of a <a href="/facts/Catalogue_of_Triangle_Cubics/FCfTztaK">catalogue</a> of <a href="/facts/Triangle_cubic/FSU8tXAb">triangle cubics</a> with detailed descriptions of several properties of more than 1200 triangle cubics. The open access journal Forum Geometricorum founded by Paul Yiu of <a href="/facts/Florida_Atlantic_University/KnuljtHR">Florida Atlantic University</a> in 2001 also provided a tremendous impetus in furthering this new found enthusiasm for triangle geometry. Unfortunately, since 2019, the journal is not accepting submissions although back issues are still available online.

Modern triangle geometry open-in-new

Modern triangle geometry