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Pedoe's Inequality
In geometry, Pedoe's inequality (also Neuberg–Pedoe inequality), named after Daniel Pedoe (1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if ''a'', ''b'', and ''c'' are the lengths of the sides of a triangle with area ''ƒ'', and ''A'', ''B'', and ''C'' are the lengths of the sides of a triangle with area ''F'', then :A^2(b^2+c^2-a^2)+B^2(a^2+c^2-b^2)+C^2(a^2+b^2-c^2)\geq 16Ff,\, with equality if and only if the two triangles are similarity (geometry), similar with pairs of corresponding sides (''A, a''), (''B, b''), and (''C, c''). The expression on the left is not only symmetric under any of the six permutations of the set of pairs, but also—perhaps not so obviously—remains the same if ''a'' is interchanged with ''A'' and ''b'' with ''B'' and ''c'' with ''C''. In other words, it is a symmetric function of the pair of triangles. Pedoe's inequality is a generalization of Weitzenböck's inequality, which is the case in which one o ... [...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] |
Daniel Pedoe
Dan Pedoe (29 October 1910, London – 27 October 1998, St Paul, Minnesota, USA) was an English-born mathematician and geometer with a career spanning more than sixty years. In the course of his life he wrote approximately fifty research and expository papers in geometry. He is also the author of various core books on mathematics and geometry some of which have remained in print for decades and been translated into several languages. These books include the three-volume ''Methods of Algebraic Geometry'' (which he wrote in collaboration with W. V. D. Hodge), ''The Gentle Art of Mathematics'', ''Circles: A Mathematical View'', ''Geometry and the Visual Arts'' and most recently ''Japanese Temple Geometry Problems: San Gaku'' (with Hidetoshi Fukagawa). Early life Daniel Pedoe was born in London in 1910, the youngest of thirteen children of Szmul Abramski, a Jewish immigrant from Poland who found himself in London in the 1890s: he had boarded a cattleboat not knowing whether it was b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Jean Baptiste Neuberg
Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926) was a Luxembourger mathematician who worked primarily in geometry. Biography Neuberg was born on 30 October 1840 in Luxembourg City, Luxembourg. He first studied at a local school, the Athénée de Luxembourg, then progressed to Ghent University, studying at the École normale des Sciences of the science faculty. After graduation, Neuberg taught at several institutions. Between 1862 and 1865, he taught at the École Normale de Nivelle. For the next sixteen years, he taught at the Athénée Royal d'Arlon, though he also taught at the École Normale at Bruges from 1868 onwards. Retrieved on 2008-09-16. Neuberg switched from his previous two schools to the Athénée Royal de Liège in 1878. He became an extraordinary professor in the university in the same city in 1884, and was promoted to ordinary professor in 1887. He held this latter position until his retirement in 1910. A year after his retirement, he was elect ... [...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] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similarity (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or reducing), possibly with additional translation (geometry), translation, rotation (mathematics), rotation and reflection (mathematics), reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruence (geometry), congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. If two angles of a triangle h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corresponding Sides
In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has sequential sides , , , , and and the other has sequential sides , , , , and , and if and are corresponding sides, then side (adjacent to ) must correspond to either or (both adjacent to ). If and correspond to each other, then corresponds to , corresponds to , and corresponds to ; hence the th element of the sequence corresponds to the th element of the sequence for On the other hand, if in addition to corresponding to we have corresponding to , then the th element of corresponds to the th element of the reverse sequence . Congruence tests look for all pairs of corresponding sides to be equal in length, though except in the case of the trian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weitzenböck's Inequality
In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt\, \Delta. Equality occurs if and only if the triangle is equilateral. Pedoe's inequality is a generalization of Weitzenböck's inequality. The Hadwiger–Finsler inequality is a strengthened version of Weitzenböck's inequality. Geometric interpretation and proof Rewriting the inequality above allows for a more concrete geometric interpretation, which in turn provides an immediate proof. : \fraca^2 + \fracb^2 + \fracc^2 \geq 3\, \Delta. Now the summands on the left side are the areas of equilateral triangles erected over the sides of the original triangle and hence the inequation states that the sum of areas of the equilateral triangles is always greater than or equal to threefold the area of the original triangle. : \Delta_a + \Delta_b + \Delta_c \geq 3\, \Delta. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heron's Formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work ''Metrica'', though it was probably known centuries earlier. Example Let be the triangle with sides , and . This triangle’s semiperimeter is :s=\frac=\frac=16 and so the area is : \begin A &= \sqrt = \sqrt\\ &= \sqrt = \sqrt = 24. \end In this example, the side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers. Alternate expressions Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways, :\begin A &=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Schwarz Inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version. Statement of the inequality The Cauchy–Schwarz inequality states that for all vectors \mathbf and \mathbf of an inner product space it is true that where \langle \cdot, \cdot \rangle is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a norm, called the or , where the norm of a vector \mathbf is denoted and defined by: \, \mathbf\, := \sqrt so that this norm and the inner product are related by the defining condition \, \mathbf\, ^2 = \langle \mathbf, \mathbf \rangle, where \langle \mathbf, \mathbf \rangle is always a non-negative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Triangle Inequalities
In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of the form "less than", "less than or equal to", "greater than", or "greater than or equal to". The parameters in a triangle inequality can be the side lengths, the semiperimeter, the angle measures, the values of trigonometric functions of those angles, the area of the triangle, the medians of the sides, the altitudes, the lengths of the internal angle bisectors from each angle to the opposite side, the perpendicular bisectors of the sides, the distance from an arbitrary point to another point, the inradius, the exradii, the circumradius, and/or other quantities. Unless otherwise specified, this article deals with triangles in the Euclidean plane. Main parameters and notation The parameters most commonly appearing in triangle inequalities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
