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  "body": "\n## Summary\n\nA similarity transformation of Euclidean space is a bijection that multiplies all distances by the same positive factor, its ratio.\nSimilarities are the compositions of [[isometries|02HPP5VCZH]] and [[homotheties|0ZG1PNEP35]], and they preserve angles, shapes and ratios of lengths.\nTwo figures are similar when a similarity maps one onto the other.\nThe theory of similar figures, such as similar triangles, rests on this notion.\n\n## Explanations\n\n### L1 human\n\nA similarity is a change of scale followed by a rigid motion.\nPhotographs enlarged or reduced, maps drawn at a scale, and models of buildings all use similarities.\nThe map that doubles every coordinate, $(x, y) \\mapsto (2x, 2y)$, is a similarity of ratio $2$: every distance doubles.\nAngles do not change, so a triangle is mapped to a triangle with the same angles.\n",
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