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Isoperimetric Point
In geometry, the isoperimetric point is a triangle center — a special point associated with a plane triangle. The term was originally introduced by G.R. Veldkamp in a paper published in the American Mathematical Monthly in 1985 to denote a point in the plane of a triangle having the property that the triangles have isoperimeters, that is, having the property that \begin & \overline + \overline + \overline, \\ =\ & \overline + \overline + \overline, \\ =\ & \overline + \overline + \overline. \end Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions. The isoperimetric point of in the sense of Veldkamp, if it exists, has the following trilinear coordinates. \sec\tfrac \cos\tfrac \cos\tfrac - 1 \ : \ \sec\tfrac \cos\tfrac \cos\tfrac - 1 \ : \ \sec\tfrac \cos\tfrac \cos\tfrac - 1 Given any triangle one can associate with it a point having trilinear coordinates as given above. This point is a triangle center and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Triangle Center
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. Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations. In other words, for any triangle and any similarity transformation (such as a rotation, reflection, dilation, or translation), the center of the transformed triangle is the same point as the transformed center of the original triangle. This invariance is the defining property of a triangle center. It rules out other well-known points such as the Brocard points which are not invariant under reflection and so fail to qualify as triangle centers. For an equilateral triangle, all triangle centers coincide at its centroid. However the triangle centers generally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... is Vadim Ponomarenko ( San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Trilinear Coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Clark Kimberling
Clark Kimberling (born November 7, 1942, in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer sequences, and hymnology. Kimberling received his PhD in mathematics in 1970 from the Illinois Institute of Technology, under the supervision of Abe Sklar. Since at least 1994, he has maintained a list of triangle centers and their properties. In its current on-line form, the Encyclopedia of Triangle Centers, this list comprises tens of thousands of entries. He has contributed to ''The Hymn'', the journal of the Hymn Society in the United States and Canada; and in the '' Canterbury Dictionary of Hymnology''. Kimberling's golden triangle Robert C. Schoen has defined a "golden triangle" as a triangle with two of its sides in the golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Encyclopedia Of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. This resource is hosted at the University of Evansville The University of Evansville (UE) is a private university in Evansville, Indiana. It was founded in 1854 as Carnegie Hall of Moores Hill College, Moores Hill College. The university operates a satellite center, Harlaxton Manor, Harlaxton College .... It started from a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles by Professor Clark Kimberling. , the list identifies 68,547 triangle centers and is managed cooperatively by an international team of geometry researchers. The encyclopedia is integrated into Geogebra. Each point in the list is identified by an index number of the form ''X''(''n'') —for example, ''X''(1) is the incenter. The information recorded about each point includes its trilinear and barycentric coordinates and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Emile Lemoine
Emile or Émile may refer to: * Émile (novel) (1827), autobiographical novel based on Émile de Girardin's early life * Emile, Canadian film made in 2003 by Carl Bessai * '' Emile: or, On Education'' (1762) by Jean-Jacques Rousseau, a treatise on education; full title ''Émile ou de l'education'' People * Emile (producer), American hip hop producer Emile Haynie * Emil (given name), includes people and characters with given name Emile or Émile * Barbara Emile, British television producer * Chris Emile, American dancer * Jonathan Emile, stage name of Jamaican-Canadian singer, rapper and record producer Jonathan Whyte Potter-Mäl (born 1986) * Yonan Emile Yonan Emile was an Iraqi basketball player. He competed in the men's tournament at the 1948 Summer Olympics The 1948 Summer Olympics, officially the Games of the XIV Olympiad and officially branded as London 1948, were an international mul ..., Iraqi Olympic basketball player * Emile Witbooi. South African soccer pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isoperimetric Point 02
In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. ''Isoperimetric'' literally means "having the same perimeter". Specifically, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that :4\pi A \le L^2, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary (topology), boundary has a specified length. The closely related ''Dido's problem'' asks for a region of the maximal area bounded by a straight line and a curvilinear arc (geometry), arc whose endpoints belong to that line. It is named after Dido (Queen of Carthage), Dido, the legendary founder and first queen of Carthage. The solution to the isoperimetric problem is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Incenter
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. Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one of the four that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers.. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gergonne Point
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
