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Fuhrmann Circle
__notoc__ In geometry, the Fuhrmann circle of a triangle, named after the German Wilhelm Fuhrmann (1833–1904), is the circle that has as a diameter the line segment between the orthocenter H and the Nagel point N. This circle is identical with the circumcircle of the Fuhrmann triangle. The radius of the Fuhrmann circle of a triangle with sides ''a'', ''b'', and ''c'' and circumradius ''R'' is : R\sqrt, which is also the distance between the circumcenter and incenter. Aside from the orthocenter the Fuhrmann circle intersects each altitude of the triangle in one additional point. Those points all have the distance 2r from their associated vertices of the triangle. Here r denotes the radius of the triangles incircle.Ross Honsberger: ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry''. MAA, 1995, pp49-52/ref> Notes Further reading *Nguyen Thanh Dung"The Feuerbach Point and the Fuhrmann Triangle" ''Forum Geometricorum'', Volume 16 (2016), pp. 299–311. * J. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuhrmann Circle
__notoc__ In geometry, the Fuhrmann circle of a triangle, named after the German Wilhelm Fuhrmann (1833–1904), is the circle that has as a diameter the line segment between the orthocenter H and the Nagel point N. This circle is identical with the circumcircle of the Fuhrmann triangle. The radius of the Fuhrmann circle of a triangle with sides ''a'', ''b'', and ''c'' and circumradius ''R'' is : R\sqrt, which is also the distance between the circumcenter and incenter. Aside from the orthocenter the Fuhrmann circle intersects each altitude of the triangle in one additional point. Those points all have the distance 2r from their associated vertices of the triangle. Here r denotes the radius of the triangles incircle.Ross Honsberger: ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry''. MAA, 1995, pp49-52/ref> Notes Further reading *Nguyen Thanh Dung"The Feuerbach Point and the Fuhrmann Triangle" ''Forum Geometricorum'', Volume 16 (2016), pp. 299–311. * J. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuhrmann Circle Triangle
Fuhrmann or Fuhrman may refer to: Surname * Bärbel Fuhrmann (born 1940), retired German swimmer * Emma Fuhrmann (born 2001), American film actress and model * Ernst Fuhrmann (1918-1995), chairman of Porsche AG in the 1970s * Irene Fuhrmann (born 1980), Austrian former football player * Joel Fuhrman (born 1953), American physician advocating a "micronutrient-rich diet" * Mark Fuhrman (born 1952), former detective of the Los Angeles Police Department, investigator in the O.J. Simpson murder case * Louis P. Fuhrmann (1868–1931), Mayor of the City of Buffalo, New York * Manfred Fuhrmann (1925-2005), German philologist * Otto Fuhrmann (1871-1945), Swiss parasitologist who specialized in the field of helminthology * Petra Fuhrmann (1955–2019), Austrian politician * Susan Fuhrmann Susan Elizabeth Fuhrmann (born 30 July 1986), known as "the Fuhrmannator", is an Australian retired international netball player. Early life Susan Fuhrmann was born and raised in Katoomba, New Sou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Fuhrmann
Wilhelm Ferdinand Fuhrmann (28 February 1833 – 11 June 1904) was a German mathematician. The Fuhrmann circle and the Fuhrmann triangle are named after him.Roger A. Johnson: ''Advanced Euclidean Geometry''. Dover 2007, , pp. 228–229, 300 (originally published 1929 with Houghton Mifflin Company (Boston) as ''Modern Geometry''). Biography Fuhrmann was born on 28 February 1833 in Burg bei Magdeburg. Fuhrmann had shortly worked as sailor before he returned to school and attended the Altstadt Gymnasium in Königsberg, where his teachers noticed his interest and talent in mathematics and geography. He graduated in 1853 and went on to study mathematics and physics at the University of Königsberg. One of his peers later remembered him as the most talented and diligent student of his class. Fuhrmann however despite his talent did not pursue a career at the university, instead he became a math and science teacher at the Burgschule in Königsberg after his graduation. He j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length d of a diameter is also called the diameter. In this sense one speaks of diameter rather than diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. :d = 2r \qquad\text\qquad r = \frac. For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as \overline). Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a chord (geometry), chord (of that curve). In real or complex vector spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ''extended base'' of the altitude. The intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nagel Point
In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurrency of all three of the triangle's splitters. Construction Given a triangle , let be the extouch points in which the -excircle meets line , the -excircle meets line , and the -excircle meets line , respectively. The lines concur in the Nagel point of triangle . Another construction of the point is to start at and trace around triangle half its perimeter, and similarly for and . Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments are called the triangle's splitters. There exists an easy construction of the Nagel point. Starting from each vertex of a triangle, it suffices to carry twice the length of the opposite edge. We obtain three lines which concur at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuhrmann Triangle
The Fuhrmann triangle, named after Wilhelm Fuhrmann (1833–1904), is special triangle based on a given arbitrary triangle. For a given triangle \triangle ABC and its circumcircle the midpoints of the arcs over triangle sides are denoted by M_a, M_b, M_c . Those midpoints get reflected at the associated triangle sides yielding the points M^\prime_a, M^\prime_b, M^\prime_c , which forms the ''Fuhrmann triangle''. Roger A. Johnson: ''Advanced Euclidean Geometry''. Dover 2007, , pp. 228–229, 300 (originally published 1929 with Houghton Mifflin Company (Boston) as ''Modern Geometry'').Ross Honsberger: ''Episodes in Nineteenth and Twentieth Century Euclidean Geometry''. MAA, 1995, pp49-52 The circumcircle of Fuhrmann triangle is the Fuhrmann circle. Furthermore the Furhmann triangle is similar to the triangle formed by the mid arc points, that is \triangle M^\prime_c M^\prime_b M^\prime_a \sim \triangle M_a M_b M_c . For the area of the Fuhrmann triangle the following fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an obtuse triangle, the minimum bounding circle has the longest side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
