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  "body": "\n## Definition\n\nA [[projective plane|DANFNN8C4Y]] is *Desarguesian* if the conclusion of [[Desargues's theorem|5Z19Y02MME]] holds in it.\nThat is, let $ABC$ and $A'B'C'$ be triangles with $A \\ne A'$, $B \\ne B'$, $C \\ne C'$ and distinct corresponding sides.\nIf $AA'$, $BB'$, $CC'$ pass through a common point, then $AB \\cap A'B'$, $BC \\cap B'C'$, $CA \\cap C'A'$ are [[collinear|0AFNNTRTJE]].\n\n## Remarks\n\nA projective plane is Desarguesian exactly when it is isomorphic to $\\mathbb{P}^2(D)$ for some division ring $D$, and then $D$ is determined up to isomorphism.\n\nExample: $\\mathbb{P}^2(\\Q)$ is Desarguesian.\nCheck: it is $\\mathbb{P}^2(k)$ for the field $k = \\Q$, where Desargues's theorem holds.\n\nNon-example: the projective completion of the [[Moulton plane|3V9GYYXM1H]] is not Desarguesian, since it contains two triangles in perspective from a point whose corresponding sides meet in three non-collinear points.\n\nBy Wedderburn's little theorem every finite division ring is a field, so the finite Desarguesian planes are the planes $\\mathbb{P}^2(\\F_q)$, of order $q$ a prime power.\nEvery Desarguesian plane is a [[Moufang plane|7F4HSNJJH8]].\n",
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