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  "body": "\n## Statement\n\nLet $P$ be a finite set with $n$ elements and let $L$ be a family of subsets of $P$, called lines, such that:\n\n1. every line has at least two points;\n2. any two distinct points lie on exactly one line;\n3. no line contains all points of $P$.\n\nThen $\\abs{L} \\ge n$.\nMoreover $\\abs{L} = n$ if and only if $(P, L)$ is either a [[projective plane|DANFNN8C4Y]] or a near-pencil.\nA near-pencil consists of one line with $n - 1$ points and the $n - 1$ lines of two points that join the remaining point to them.\n\n## Remarks\n\nFor a point $p$ let $r_p$ be the number of lines through $p$, and for a line $\\ell$ let $k_\\ell$ be its number of points.\nIf $p$ is not on $\\ell$, the lines joining $p$ to the points of $\\ell$ are distinct, so $r_p \\ge k_\\ell$.\nWrite $b = \\abs{L}$ and sum $1/(n(b - r_p))$ and $1/(b(n - k_\\ell))$ over all pairs $(p, \\ell)$ with $p \\notin \\ell$: each sum equals $1$.\nIf $b \u003c n$, then $n r_p \u003e b k_\\ell$ for each such pair, so each term of the first sum is larger than the matching term of the second, a contradiction.\nA closer look at the case $b = n$ gives the description of equality.\n\nHypothesis 3 is needed: a single line through all $n \\ge 3$ points satisfies clauses 1 and 2 and has only one line.\n\nA consequence: $n$ points of the real plane, not all on one line, determine at least $n$ distinct lines through two or more of them.\nThe theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős.\n\n## Explanations\n\n### L1 human\n\nTake a finite set of points in which every two points lie on a unique line, and not all points are on one line.\nThe theorem says there are at least as many lines as points.\nThere are two ways to reach equality.\nOne is a projective plane, such as the Fano plane with seven points and seven lines.\nThe other is the near-pencil: a line with all points but one, plus the lines through the last point.\n\n### L2 human\n\nThe statement lives in a [[linear space|CX07CWPCDX]], a partial linear space in which any two points are joined.\nIt is close in spirit to Fisher's inequality for block designs.\nApplied to points in the real plane, it gives a lower bound for the number of lines they determine, a question related to the Sylvester–Gallai theorem.\nThe equality case shows how rigid projective planes are among all finite linear spaces.\n",
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