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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mass point geometry ： ウィキペディア英語版 Mass point geometry Mass point geometry, colloquially known as mass points, is a geometry problem-solving technique which applies the physical principle of the center of mass to geometry problems involving triangles and intersecting cevians.〔Rhoad, R., Milauskas, G., and Whipple, R. ''Geometry for Enjoyment and Challenge''. McDougal, Littell & Company, 1991.〕 All problems that can be solved using mass point geometry can also be solved using either similar triangles, vectors, or area ratios,〔http://mathcircle.berkeley.edu/archivedocs/2007_2008/lectures/0708lecturesps/MassPointsBMC07.ps〕 but mass point geometry is far quicker than those methods〔http://www.artofproblemsolving.com/Wiki/index.php/Mass_Point_Geometry〕 and thus is used more often on math competitions in which time is an important factor. Though modern mass point geometry was developed in the 1960s by New York high school students,〔Rhoad, R., Milauskas, G., and Whipple, R. ''Geometry for Enjoyment and Challenge''. McDougal, Littell & Company, 1991〕 the concept has been found to have been used as early as 1827 by August Ferdinand Möbius in his theory of homogeneous coordinates.〔D. Pedoe ''Notes on the History of Geometrical Ideas I: Homogeneous Coordinates''. Math Magazine (1975), 215-217.〕 == Definitions == The theory of mass points is rigorously defined according to the following definitions:〔H. S. M. Coxeter, ''Introduction to Geometry'', pp. 216-221, John Wiley & Sons, Inc. 1969〕 * Mass Point - A mass point is a pair $(m,\; P)$, also written as $mP$, including a mass, $m$, and an ordinary point, $P$ on a plane. * Coincidence - We say that two points $mP$ and $nQ$ coincide if and only if $m\; =\; n$ and $P\; =\; Q$. * Addition - The sum of two mass points $mP$ and $nQ$ has mass $m\; +\; n$ and point $R$ where $R$ is the point on $PQ$ such that $PR:RQ\; =\; n:m$. In other words, $R$ is the fulcrum point that perfectly balances the points $P$ and $Q$. An example of mass point addition is shown at right. Mass point addition is closed, idempotent, commutative, and associative. * Scalar Multiplication - Given a mass point $mP$ and a positive real scalar $k$, we define multiplication to be $k(m,\; P)\; =\; (km,\; P)$. Mass point scalar multiplication is distributive over mass point addition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mass point geometry」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.