{
  "aliases": [
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  "body": "\n## Statement\n\nLet $S$ be a point of the Euclidean plane, and let $\\ell_1$ and $\\ell_2$ be two distinct [[lines|0MP9BCF6DP]] through $S$. Let $A$ and $B$ be points of $\\ell_1$, and let $C$ and $D$ be points of $\\ell_2$, all different from $S$. Let $|XY|$ denote the [[distance|0NPPGHQNXQ]] between the points $X$ and $Y$.\nAssume that the line $AC$ is parallel to the line $BD$. Then\n$$\\frac{|SA|}{|SB|} = \\frac{|SC|}{|SD|} = \\frac{|AC|}{|BD|}.$$\n\n## Remarks\n\nIf also $A \\neq B$ and $A$ and $B$ are on the same ray from $S$, then the theorem also gives $\\frac{|SA|}{|AB|} = \\frac{|SC|}{|CD|}$.\nA converse holds for two rays from $S$: if $A, B$ are on one ray, $C, D$ on the other, and $\\frac{|SA|}{|AB|} = \\frac{|SC|}{|CD|}$, then $AC$ is parallel to $BD$.\nThe theorem says that the triangles $SAC$ and $SBD$ are [[similar|0YF6QXXWB5]]. The [[homothety|0ZG1PNEP35]] with the centre $S$ that sends $A$ to $B$ also sends $C$ to $D$.\n\n## History\n\nTradition attributes the theorem to Thales of Miletus. He may have used it to find the heights of the pyramids of Egypt.\n\n## Explanations\n\n### L1 human\n\nDraw two lines from one point, and cut them with two parallel lines. The parallel lines cut both lines in the same proportions. This is how you can find the height of a tree from the length of its shadow and the shadow of a stick.\n",
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