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  "body": "\n## Definition\n\nLet $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean space $\\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].\nThe triangles are similar, written $ABC \\sim A'B'C'$, if there is a real number $r \u003e 0$ such that\n$$d(A', B') = r \\, d(A, B), \\quad d(B', C') = r \\, d(B, C) \\quad \\text{and} \\quad d(C', A') = r \\, d(C, A).$$\nThe number $r$ is the ratio of the similarity.\n\n## Remarks\n\nSimilar triangles have equal corresponding angles. Conversely, the [[angle-angle criterion|0NZVJYZVTZ]] says that two pairs of equal angles are enough for similarity.\n[[Congruent triangles|093EDENVA1]] are the similar triangles with the ratio $r = 1$.\nTwo triangles are similar if and only if a [[homothety|0ZG1PNEP35]] followed by an isometry sends one onto the other, vertex by vertex.\nAny two equilateral triangles are similar.\n\n## Explanations\n\n### L1 human\n\nA photo of a triangle, enlarged or reduced, gives a similar triangle. All sides change by the same factor, and the angles stay the same. For example, the triangles with sides $3, 4, 5$ and $6, 8, 10$ are similar with the ratio $2$.\n",
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