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Barrow's Inequality
In geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain points on the sides of the triangle. It is named after David Francis Barrow. Statement Let ''P'' be an arbitrary point inside the triangle ''ABC''. From ''P'' and ''ABC'', define ''U'', ''V'', and ''W'' as the points where the angle bisectors of ''BPC'', ''CPA'', and ''APB'' intersect the sides ''BC'', ''CA'', ''AB'', respectively. Then Barrow's inequality states that : PA+PB+PC\geq 2(PU+PV+PW),\, with equality holding only in the case of an equilateral triangle and ''P'' is the center of the triangle. Generalisation Barrow's inequality can be extended to convex polygons. For a convex polygon with vertices A_1,A_2,\ldots ,A_n let P be an inner point and Q_1, Q_2,\ldots ,Q_n the intersections of the angle bisectors of \angle A_1PA_2,\ldots,\angle A_PA_n,\angle A_nPA_1 with the associated polygon sides A_1A_2,\ldots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrow Inequality
In geometry, Barrow's inequality is an Inequality (mathematics), inequality relating the Euclidean distance, distances between an arbitrary point within a triangle, the vertices of the triangle, and certain points on the sides of the triangle. It is named after David Francis Barrow. Statement Let ''P'' be an arbitrary point inside the triangle ''ABC''. From ''P'' and ''ABC'', define ''U'', ''V'', and ''W'' as the points where the angle bisectors of ''BPC'', ''CPA'', and ''APB'' intersect the sides ''BC'', ''CA'', ''AB'', respectively. Then Barrow's inequality states that : PA+PB+PC\geq 2(PU+PV+PW),\, with equality holding only in the case of an equilateral triangle and ''P'' is the center of the triangle. Generalisation Barrow's inequality can be extended to convex polygons. For a convex polygon with vertices A_1,A_2,\ldots ,A_n let P be an inner point and Q_1, Q_2,\ldots ,Q_n the intersections of the angle bisectors of \angle A_1PA_2,\ldots,\angle A_PA_n,\angle A_nPA_1 wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Theorem In Geometry
In geometry, Euler's theorem states that the distance ''d'' between the circumcenter and incenter of a triangle is given by d^2=R (R-2r) or equivalently \frac + \frac = \frac, where R and r denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, who published it in 1765. However, the same result was published earlier by William Chapple in 1746. From the theorem follows the Euler inequality: R \ge 2r, which holds with equality only in the equilateral case. Stronger version of the inequality A stronger version is \frac \geq \frac \geq \frac+\frac+\frac-1 \geq \frac \left(\frac+\frac+\frac \right) \geq 2, where a, b, and c are the side lengths of the triangle. Euler's theorem for the escribed circle If r_a and d_a denote respectively the radius of the escribed circle opposite to the vertex A and the distance between its center and the center of the circumscribed circl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Gazette
''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching. Its publisher is the Mathematical Association. William John Greenstreet was its editor for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors * Edward Mann Langley: 1894-1896 * Francis Sowerby Macaulay: 1896-1897 * William John Greenstreet: 1897-1930 * Alan Broadbent: 1930-1955 * Reuben Goodstein: 1956-1962 * Edwin A. Maxwell: 1962-1971 * Douglas Quadling Douglas Arthur Quadling (1926–2015) was an English mathematician, school master and educationalist who was one of the four drivers behind the School Mathematics Project (SMP) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis J
Louis may refer to: * Louis (coin) * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer * HMS Louis, HMS ''Louis'', two ships of the Royal Navy See also Derived or associated terms * Lewis (other) * Louie (other) * Luis (other) * Louise (other) * Louisville (other) * Louis Cruise Lines * Louis dressing, for salad * Louis Quinze, design style Associated names * * Chlodwig, the origin of the name Ludwig, which is translated to English as "Louis" * Ladislav and László - names sometimes erroneously associated with "Louis" * Ludovic, Ludwig (other), Ludwig, Ludwick, Ludwik, names sometimes translated to English as "Louis" {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annali Di Matematica Pura Ed Applicata
The ''Annali di Matematica Pura ed Applicata'' (Annals of Pure and Applied Mathematics) is a bimonthly peer-reviewed scientific journal covering all aspects of pure and applied mathematics. The journal was established in 1850 under the title of ''Annali di scienze matematiche e fisiche'' (Annals of Mathematics and Physics), and changed to its current title in 1858: it was the first Italian periodical devoted to mathematics and written in Italian.. The founding editors-in-chief were Barnaba Tortolini and Francesco Brioschi. It is currently published by Springer Science+Business Media and the editor-in-chief is Graziano Gentili (University of Florence). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' 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. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Mordell Inequality
In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ''ABC'' and point ''P'' inside ''ABC'', the sum of the distances from ''P'' to the sides is less than or equal to half of the sum of the distances from ''P'' to the vertices. It is named after Paul Erdős and Louis Mordell. posed the problem of proving the inequality; a proof was provided two years later by . This solution was however not very elementary. Subsequent simpler proofs were then found by , , and . Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from ''P'' to the sides are replaced by the distances from ''P'' to the points where the angle bisectors of ∠''APB'', ∠''BPC'', and ∠''CPA'' cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices. Statement Let P be an arbitrary point P inside a given triangle ABC, and let PL, PM, and PN be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cos ... [...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] |
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] |
Angle Bisector
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through the midpoint of a given segment) and the ''angle bisector'' (a line that passes through the apex of an angle, that divides it into two equal angles). In three-dimensional space, bisection is usually done by a plane, also called the ''bisector'' or ''bisecting plane''. Perpendicular line segment bisector Definition *The perpendicular bisector of a line segment is a line, which meets the segment at its midpoint perpendicularly. The Horizontal intersector of a segment AB also has the property that each of its points X is equidistant from the segment's endpoints: (D)\quad , XA, = , XB, . The proof follows from and Pythagoras' theorem: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^2=, XB, ^2 \; . Property (D) is usually used for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
