Modern triangle geometry, also known as new triangle geometry, is the body of mathematical knowledge relating to the properties of a triangle that was discovered and developed roughly from the beginning of the last quarter of the nineteenth century onward. It is a recognized branch of Euclidean geometry.
Historical background
Triangles and their properties have been subjects of investigation since at least the time of Euclid. Euclid's Elements describes four special points associated with a triangle: the centroid, incenter, circumcenter, and orthocenter. Although mathematicians such as Pascal and Ceva (seventeenth century), Euler (eighteenth century), and Feuerbach (nineteenth century) made important discoveries regarding triangle properties, the publication in 1873 of a paper by Émile Lemoine (1840–1912) titled "On a remarkable point of the triangle" is widely regarded as the foundational event of modern triangle geometry. According to Nathan Altschiller-Court, this paper "laid the foundations...of the modern geometry of the triangle as a whole." The American Mathematical Monthly stated 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 Lemoine's paper generated a significant upsurge of interest in investigating triangle properties during the last quarter of the nineteenth century and the early years of the twentieth century. A hundred-page article on triangle geometry in Klein's Encyclopedia of Mathematical Sciences, published in 1914, attests to this interest.
Early development and decline
In its early period, "new triangle geometry" referred primarily to the set of notable objects associated with a triangle, such as the Lemoine point, Lemoine circle, Brocard circle, and the Lemoine line. Later, the theory of correspondences—an offshoot of the theory of geometric transformations—was developed to provide coherence to the various isolated results.
Interest in triangle geometry subsequently declined. In his Development of Mathematics (1940), Eric Temple Bell wrote: "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." Philip Davis has suggested several reasons for this decline, including the perception of the subject as elementary and of low professional status, the exhaustion of its methodological possibilities, the visual complexity of its deeper results, the downgrading of visual approaches in favor of algebraic ones, a dearth of connections to other fields, and competition from other visually rich topics such as tessellations, fractals, and graph theory.
Revival and contemporary developments
A revival of interest occurred with the advent of the modern electronic computer. Triangle geometry again became an active area of research. Notable developments include the formulation of the concept of a "triangle center" by Clark Kimberling in 1994, and the compilation of the Encyclopedia of Triangle Centers, which lists nearly 70,000 triangle centers and their properties. Bernard Gibert's Catalogue of Triangle Cubics provides detailed descriptions of more than 1,200 triangle cubics. The open-access journal Forum Geometricorum, founded by Paul Yiu of Florida Atlantic University in 2001, also contributed to this revival, although it stopped accepting submissions in 2019.
Key concepts and objects
Modern triangle geometry encompasses numerous specific objects and concepts, including:
- Lemoine point: The point of concurrency of the three symmedians of a triangle, also called the symmedian point or Grebe point. It is the isogonal conjugate of the centroid.
- Lemoine circles: The first and second Lemoine circles, the latter also known as the cosine circle.
- Lemoine axis: The axis of perspectivity between a triangle and its tangential triangle.
- Poristic triangles: Triangles sharing the same incircle and circumcircle.
- Simson line: For a point on the circumcircle, the line through the feet of the perpendiculars from that point to the sides of the triangle.
- Pedal and antipedal triangles: Triangles formed by perpendicular projections from a point to the sides of a reference triangle.
- Orthopole: The point of concurrence of lines associated with perpendiculars from triangle vertices to a given line.
- Brocard points: Two special points within a triangle associated with equal angles, along with the Brocard angle.
- Triangle centers: Points defined by functions of the side lengths, unifying many special points associated with a triangle.
- Central lines: Lines in the plane of a triangle whose trilinear equations are defined in terms of triangle center functions.
- Triangle conics: Conics associated with a reference triangle, such as the Steiner ellipse, Kiepert hyperbola, and Artzt parabolas.
- Triangle cubics: Cubic curves arising in the study of triangles, such as the Neuberg cubic, McCay cubic, and Tucker cubic.
Computational contributions
Computers have played a significant role in generating new results in triangle geometry. The computer program "Discoverer" has produced new theorems, including results on isogonal conjugates, the Kiepert hyperbola, and generalized Lester circles. A web portal dedicated to a computer-discovered encyclopedia of Euclidean geometry is maintained by Sava Grozdev, Hiroshi Okumura, and Deko Dekov.
References
Key works in the field include William Gallatly's The Modern Geometry of the Triangle (1910), Roger A. Johnson's Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle (1929, later published as Advanced Euclidean Geometry), Ross Honsberger's Episodes in Nineteenth and Twentieth Century Euclidean Geometry (1995), and Nathan Altschiller-Court's College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle.
