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  "body": "\n## Definition\n\nLet $K$ be a field, and let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space $V$. Let $c$ be a point of $A$ and let $k$ be a nonzero element of $K$.\nThe homothety with the centre $c$ and the ratio $k$ is the map $h : A \\to A$ with\n$$h(p) = c + k (p - c) \\quad \\text{for each point } p \\text{ of } A.$$\n\n## Remarks\n\nA homothety is an [[affine map|0JF0TVYGWT]] with the linear part $v \\mapsto k v$. It fixes its centre, and it sends each line to a parallel line.\nThe ratio $k = 1$ gives the identity. The ratio $k = -1$ gives the point reflection in $c$.\nIn a Euclidean space, a homothety with the ratio $k$ multiplies each distance by $|k|$, each area by $k^2$ and each volume by $|k|^3$.\nMathlib's `AffineMap.homothety` allows each $k$, also $k = 0$, which gives the constant map with the value $c$.\n\n## Explanations\n\n### L1 human\n\nA homothety is a zoom centred at a point. With $k = 2$, each point moves to twice its distance from the centre, on the same ray. A triangle becomes a triangle with sides twice as long, and the same angles.\n",
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