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  "body": "\n## Statement\n\nLet $\\Pi$ be a finite [[projective plane|DANFNN8C4Y]] of [[order|K2YVMTGTBB]] $n$.\nThen every point of $\\Pi$ lies on exactly $n + 1$ lines, and $\\Pi$ has exactly $n^2 + n + 1$ points and exactly $n^2 + n + 1$ lines.\n\n## Remarks\n\nFix a point $P$.\nIt lies on $n + 1$ lines, and each of them has $n$ points other than $P$.\nEvery other point $Q$ lies on exactly one line through $P$, namely $PQ$, so the number of points is $1 + (n + 1) n = n^2 + n + 1$.\nThe count of lines follows by the same argument in the dual plane, which is again a projective plane of order $n$.\n\nExample: the [[Fano plane|QQ8GW1ASZM]] has order $2$ and $4 + 2 + 1 = 7$ points.\nThe plane $PG(2, q)$ has $(q^3 - 1)/(q - 1) = q^2 + q + 1$ points, as it should.\nIn Mathlib the count of points is `Configuration.ProjectivePlane.card_points`.\n\n## Explanations\n\n### L1 human\n\nOnce one knows how many points lie on a line of a finite projective plane, one knows the size of the whole plane.\nWith $n + 1$ points on each line, there are $n^2 + n + 1$ points and the same number of lines.\nFor order $2$ this gives the $7$ points of the Fano plane, and for order $3$ it gives $13$.\n\n### L2 human\n\nThe proof is a standard double count from a single point, and the same method gives the counts of flags and of triangles.\nIt shows that a projective plane of order $n$ is a symmetric block design with parameters $(n^2 + n + 1, n + 1, 1)$, a [[Steiner system|7BPSC4T4M3]] $S(2, n + 1, n^2 + n + 1)$.\nThis turns existence questions for planes into questions about designs, which is the setting of the Bruck–Ryser–Chowla theorem.\n",
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