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  "body": "\n## Statement\n\nLet $S$ be an infinite set of points in the [[Euclidean plane|0XW7V9ZZ05]].\nAssume that the [[Euclidean distance|0NPPGHQNXQ]] between any two points of $S$ is an integer.\nThen all points of $S$ lie on one line.\n\n## Remarks\n\nSuppose that $S$ contains three points $A, B, C$ that are not [[collinear|0AFNNTRTJE]].\nFor any other point $X$ of $S$, the triangle inequality gives $\\abs{\\,\\norm{X - A} - \\norm{X - B}\\,} \\le \\norm{A - B}$, and this difference is an integer.\nSo $X$ lies on one of finitely many [[hyperbolas|0WN5G117YC]] with foci $A$ and $B$, counting degenerate ones such as the perpendicular bisector and parts of the line $AB$.\nIn the same way $X$ lies on one of finitely many hyperbolas with foci $A$ and $C$, and two such curves with different foci meet in at most $4$ points.\nSo $S$ is finite, a contradiction.\nThe hypothesis that $S$ is infinite is needed: the vertices of a $3$-$4$-$5$ right triangle are three non-collinear points with integer distances.\nInteger distances cannot be weakened to rational ones: there are infinite non-collinear sets with all distances rational, for example dense subsets of a circle.\n\n## Explanations\n\n### L1 human\n\nIf infinitely many points in the plane have whole-number distances between every two of them, then they all sit on one straight line.\nOff a line, integer distances are so restrictive that only finitely many points can fit.\n\n### L2 human\n\nNorman Anning and Paul Erdős proved the theorem in 1945.\nThe proof is a short argument with hyperbolas, using the definition of a hyperbola as the set of points with a fixed difference of distances to two foci.\nIn contrast, for every $n$ there are $n$ points in the plane, no three on a line, with all distances integers; one can take suitable points on a circle.\n",
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  "name": "Erdős–Anning theorem",
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