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In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
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.
In geometry, the GEOS circle is derived from the intersection of four lines that are associated with a generalized triangle: the Euler line, the Soddy line, the orthic axis and the Gergonne line. Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is orthogonal to the Gergonne line.
In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the center of mass of the uniform-density boundary of triangle. The Spieker center is also the point where all three cleavers of the triangle intersect each other.
In geometry, a set of Johnson circles comprises three circles of equal radius r sharing one common point of intersection H. In such a configuration the circles usually have a total of four intersections : the common point H that they all share, and for each of the three pairs of circles one more intersection point. If any two of the circles happen to osculate, they only have H as a common point, and it will then be considered that H be their 2-wise intersection as well; if they should coincide we declare their 2-wise intersection be the point diametrically opposite H. The three 2-wise intersection points define the reference triangle of the figure. The concept is named after Roger Arthur Johnson.
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.
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In geometry, the Exeter point is a special point associated with a plane triangle. It is a triangle center and is designated as X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986. This is one of the recent triangle centers, unlike the classical triangle centers like centroid, incenter, and Steiner point.
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In geometry, the equal parallelians point is a special point associated with a plane triangle. It is a triangle center and it is denoted by X(192) in Clark Kimberling's Encyclopedia of Triangle Centers. There is a reference to this point in one of Peter Yff's notebooks, written in 1961.
In plane geometry, the Morley centers are two special points associated with a triangle. Both of them are triangle centers. One of them called first Morley center is designated as X(356) in Clark Kimberling's Encyclopedia of Triangle Centers, while the other point called second Morley center is designated as X(357). The two points are also related to Morley's trisector theorem which was discovered by Frank Morley in around 1899.
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994.
In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle.
In Euclidean geometry, the Vecten points are two triangle centers, points associated with any triangle. They may be found by constructing three squares on the sides of the triangle, connecting each square centre by a line to the opposite triangle point, and finding the point where these three lines meet. The outer and inner Vecten points differ according to whether the squares are extended outward from the triangle sides, or inward.
In Euclidean geometry, the Clawson point is a special point in a triangle defined by the trilinear coordinates tan α : tan β : tan γ, where α, β, γ are the interior angles at the triangle vertices A, B, C. It is named after John Wentworth Clawson, who published it 1925 in the American Mathematical Monthly. It is denoted X(19) in Clark Kimberling's Encyclopedia of Triangle Centers.
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
- ↑ Triangle Centers and Central Triangles. Congressus numerantium. Utilitas Mathematica Publishing. 1998.
- ↑ Kimberling, Clark. "Part 31: Centers X(52001) - X(54000)". Encyclopedia of Triangle Centers. Retrieved February 6, 2024.
- ↑ Kimberling, Clark. "Thanks". Encyclopedia of Triangle Centers. Retrieved February 6, 2024.
- ↑ "TriangleCenter_Command". Geogebra.
- ↑ Weisstein, Eric W. "Kimberling Center". MathWorld--A Wolfram Web Resource.