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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 Similarity (geometry) ： ウィキペディア英語版 Similarity (geometry) Two geometrical objects are called similar if they both 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 (enlarging or shrinking), possibly with additional translation, rotation and 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 congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. 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 have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar. ==Similar triangles== In geometry two triangles, $\backslash triangle\; ABC$ and $\backslash triangle\; A\text{'}B\text{'}C\text{'}$, are similar if and only if corresponding angles have the same measure : this implies that they are similar if and only if the lengths of corresponding sides are proportional. It can be shown that two triangles having congruent angles (''equiangular triangles'') are similar, that is, the corresponding sides can be proved to be proportional. This is known as the ''AAA similarity theorem''.〔. This is also proved in Euclid's Elements, Book VI, Proposition 4.〕 Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.〔For instance, and 〕 There are several statements each of which is necessary and sufficient for two triangles to be similar: 1. The triangles have two congruent angles,〔Euclid's elements Book VI Proposition 4.〕 which in Euclidean geometry implies that all their angles are congruent.〔This statement is not true in Non-euclidean geometry where the triangle angle sum is not 180 degrees.〕 That is: :If $\backslash angle\; BAC$ is equal in measure to $\backslash angle\; B\text{'}A\text{'}C\text{'}$, and $\backslash angle\; ABC$ is equal in measure to $\backslash angle\; A\text{'}B\text{'}C\text{'}$, then this implies that $\backslash angle\; ACB$ is equal in measure to $\backslash angle\; A\text{'}C\text{'}B\text{'}$ and the triangles are similar. 2. All the corresponding sides have lengths in the same ratio:〔Euclid's elements Book VI Proposition 5〕 : $=\; =$. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other. 3. Two sides have lengths in the same ratio, and the angles included between these sides have the same measure.〔Euclid's elements Book VI Proposition 6〕 For instance: : $=$ and $\backslash angle\; ABC$ is equal in measure to $\backslash angle\; A\text{'}B\text{'}C\text{'}$. This is known as the ''SAS Similarity Criterion''. When two triangles $\backslash triangle\; ABC$ and $\backslash triangle\; A\text{'}B\text{'}C\text{'}$ are similar, one writes〔Posamentier, Alfred S. and Lehmann, Ingmar. ''The Secrets of Triangles'', Prometheus Books, 2012.〕 :$\backslash triangle\; ABC\backslash sim\backslash triangle\; A\text{'}B\text{'}C\text{'}\; \backslash ,$. There are several elementary results concerning similar triangles in Euclidean geometry: * Any two equilateral triangles are similar. * Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles). * Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. * Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio.〔.〕 Given a triangle $\backslash triangle\; ABC$ and a line segment $\backslash overline$ one can, with straightedge and compass, find a point ''F'' such that $\backslash triangle\; ABC\; \backslash sim\; \backslash triangle\; DEF$. The statement that the point ''F'' satisfying this condition exists is ''Wallis's Postulate''〔Named for John Wallis (1616-1703)〕 and is logically equivalent to Euclid's Parallel Postulate. In hyperbolic geometry (where Wallis's Postulate is false) similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by G.D. Birkhoff (see Birkhoff's axioms) the SAS Similarity Criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms.〔 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「Similarity (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.