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  "body": "\n## Statement\n\nLet $S$ be a finite set of points in the [[Euclidean plane|0XW7V9ZZ05]].\nSuppose that the points of $S$ do not all lie on one line.\nThen there is a line that contains exactly two points of $S$.\n\n## Remarks\n\nA line that contains exactly two points of $S$ is called an ordinary line.\nKelly's proof considers all pairs of a point $p \\in S$ and a line $\\ell$ through two points of $S$ with $p \\notin \\ell$.\nIt takes a pair with the least distance from $p$ to $\\ell$.\nSuppose $\\ell$ contained three points of $S$.\nThen two of them would lie on the same side of the foot of the perpendicular from $p$, or one of them at the foot.\nThis gives a pair with a smaller positive distance, a contradiction; so $\\ell$ is ordinary.\nThe hypothesis that $S$ is finite is needed: the integer lattice $\\Z^2$ has no ordinary line.\nThe theorem also fails over the complex numbers.\nThe nine inflection points of a smooth complex cubic curve form a configuration in which every line through two of them contains a third.\nFor example, the four vertices of a square have ordinary lines: each side contains exactly two of the vertices.\nGreen and Tao proved that for all sufficiently large $n$, a set of $n$ points not all on a line has at least $n/2$ ordinary lines.\n\n## Explanations\n\n### L1 human\n\nScatter finitely many points in the plane, not all on one straight line.\nThen you can always find a line passing through exactly two of the points.\nThis is easy to believe but hard to prove without a clever idea.\n\n### L2 human\n\nThe Sylvester–Gallai theorem is a starting point of combinatorial incidence geometry, like the [[Szemerédi–Trotter theorem|NQBWFSC1E0]].\nA corollary is that $n$ points not all on a line determine at least $n$ distinct lines.\nIts failure over $\\C$ and over finite fields shows that it depends on the order structure of the real numbers.\n",
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