Forum Geometricorum

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<span class="mw-page-title-main">Feuerbach point</span> Point where the incircle and nine-point circle of a triangle are tangent

In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on the placement and scale of the triangle. It is listed as X(11) in Clark Kimberling's Encyclopedia of Triangle Centers, and is named after Karl Wilhelm Feuerbach.

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<span class="mw-page-title-main">Schiffler point</span> Point defined as a triangle center

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In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices. The nine-point center is listed as point X(5) in Clark Kimberling's Encyclopedia of Triangle Centers.

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<span class="mw-page-title-main">Lester's theorem</span> Several points associated with a scalene triangle lie on the same circle

In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle. The result is named after June Lester, who published it in 1997, and the circle through these points was called the Lester circle by Clark Kimberling. Lester proved the result by using the properties of complex numbers; subsequent authors have given elementary proofs, proofs using vector arithmetic, and computerized proofs. The center of the Lester circle is also a triangle center. It is the center designated as X(1116) in the Encyclopedia of Triangle Centers. Recently, Peter Moses discovered 21 other triangle centers lie on the Lester circle. The points are numbered X(15535) – X(15555) in the Encyclopedia of Triangle Centers.

<span class="mw-page-title-main">Steiner–Lehmus theorem</span> Every triangle with two angle bisectors of equal lengths is isosceles

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<span class="mw-page-title-main">Mittenpunkt</span> Triangle center: symmedian point of the triangles excentral triangle

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<span class="mw-page-title-main">Kosnita's theorem</span> Concurrency of lines connecting to certain circles associated with an arbitrary triangle

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<span class="mw-page-title-main">Van Lamoen circle</span>

In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle . It contains the circumcenters of the six triangles that are defined inside by its three medians.

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<span class="mw-page-title-main">Modern triangle geometry</span>

In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle 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 Euclid. 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." The American Mathematical Monthly, 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 Klein's Encyclopedia of Mathematical Sciences published in 1914 bears witness to this upsurge of interest in triangle geometry.

References

  1. Kimberling, Clark (2003), Geometry in Action: A Discovery Approach Using the Geometer's Sketchpad, Springer, p. 111, ISBN   9781931914024 .
  2. "Journal Information for "Forum Geometricorum. A Journal on Classical Euclidean Geometry and Related Areas"", MathSciNet , American Mathematical Society, retrieved 2015-08-26.
  3. zbMath - Forum Geometricorum - A Journal on Classical Euclidean Geometry and Related Areas, European Mathematical Society , retrieved 2020-05-28.
  4. 1 2 Suceavă, Bogdan D. (November 2020), "Letter to the Editor" (PDF), Notices of the American Mathematical Society, 67 (10): 1485