# Triangle Geometers

 Euclid's Elements and other remnants from ancient Greek times contain theorems about triangles and descriptions of four triangle centers: centroid, incenter, circumcenter, and orthocenter. Later triangle geometers include Euler, Pascal, Ceva, and Feuerbach. In 1873, Emile Lemoine presented a paper "on a remarkable point of the triangle," now known as the Lemoine point or symmedian point. This paper, writes Nathan Altshiller Court (College Geometry, page 304), "may be said to have laid the foundations...of the modern geometry of the triangle as a whole." Court also describes seminal papers by Henri Brocard and J. Neuberg and names Lemoine, Brocard, and Neuberg as the three co-founders of modern triangle geometry.

An astonishing wave of interest and publications in triangle geometry swept through the last years of the 19th century and then collapsed during the early years of the 20th.

However, many new gemstones in the fields of triangle geometry remained to be unearthed with new excavating tools, such as computers and methods from other areas of mathematics. All of this has led to the state of the art up to 1995, as described in

Philip J. Davis, "The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-history," American Mathematical Monthly 102 (1995) 204-214.

Among authors of frequently cited books in triangle geometry are the following:

Nathan Altshiller Court (1881-1968), author of Modern Geometry
Roger Arthur Johnson (1890-1954), author of Advanced Euclidean Geometry
William Gallatly (1850-1914), author of The Modern Geometry of the Triangle
John Casey (1820-1891), author of 19th century books specializing in triangles and conics
Charlotte Angas Scott (1858-1931), author of An Introductory Account of Certain Modern Ideas and Methods in Plane Analytical Geometry

Among those for whom triangle centers (or central lines, etc.) have been named are

Napoleon Bonaparte (1769-1821), as in Napoleon theorem
Giovanni Ceva (c1647-1734) as in Ceva's theorem, cevians, cevian triangle
John Wentworth Clawson (1881-1964) as in Clawson point
Leonhard Euler (1707-1783), as in Euler line
Karl Wilhelm Feuerbach (1800-1834), as in Feuerbach theorem
Joseph Diaz Gergonne (1771-1859) as in Gergonne point
Ludwig Kiepert (1846-1934) as in Kiepert hyperbola
Emile Lemoine (1840-1912) as in Lemoine point (or symmedian point)
G. de Longchamps (1842-1906) as in De Longchamps point
Gian Francesco Malfatti (1731-1807) as in Malfatti problem
Frank Morley (1860-1937) as in Morley triangle, Morley points
Christian Heinrich von Nagel (1803-1882), as in Nagel point
Joseph Jean Baptiste Neuberg (1840-1926) as in Neuberg circles
Kurt Schiffler (1896-1986) as in Schiffler point
Robert Simson (1687-1768) as in Simson line
Sir Frederick Soddy (1877-1956) as in Soddy circles
Jakob Steiner (1796-1863) as in Steiner ellipse, Steiner point