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  "body": "\n## Statement\n\nThere are constants $c_1 \u003e 0$ and $c_2 \u003e 0$ with the following property.\nLet $P$ be a finite set of $n$ points in the [[Euclidean plane|0XW7V9ZZ05]].\nSay that a [[line|0MP9BCF6DP]] is *determined* by $P$ if it contains at least two points of $P$.\nThen at least one of the following holds.\n\n1. Some line contains at least $c_1 n$ points of $P$.\n2. $P$ determines at least $c_2 n^2$ distinct lines.\n\n## Remarks\n\nThe proof counts pairs of points of $P$ according to the number of points on the line through them.\nThe [[Szemerédi–Trotter theorem|NQBWFSC1E0]] bounds the number of lines containing between $k$ and $2k$ points of $P$ by $O(n^2/k^3 + n/k)$.\nSum this bound over dyadic ranges of $k$.\nUnless some line is very rich, a constant fraction of the $\\binom{n}{2}$ pairs then lie on lines with few points.\nThose lines are then of order $n^2$ in number.\nExample: if $P$ consists of $n - 1$ [[collinear|0AFNNTRTJE]] points and one point off their line, clause 1 holds, while $P$ determines only $n$ lines.\nIf $P$ is the set of vertices of a regular $n$-gon, no line contains more than $2$ points, so clause 2 must hold; indeed all $\\binom{n}{2}$ lines are distinct.\n\n## Explanations\n\n### L1 human\n\nTake $n$ points in the plane.\nEither many of them lie on one line, or the points determine a huge number of lines, about as many as there are pairs of points.\nThere is no middle ground where the points are spread out but determine few lines.\n\n### L2 human\n\nJózsef Beck proved the theorem in 1983, and it is a standard application of the Szemerédi–Trotter incidence bound.\nA consequence is a weak form of Dirac's conjecture: some point of $P$ lies on at least $c n$ of the lines determined by $P$, unless most points are collinear.\nIt belongs to the circle of results around the [[Sylvester–Gallai theorem|T3GGJN70P2]] and the orchard-planting problem, which count lines with few points.\n",
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