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  "body": "\n## Statement\n\nLet $k$ be a field, or more generally a division ring, and work in the [[projective plane|DANFNN8C4Y]] $\\mathbb{P}^2(k)$.\nLet $A, B, C$ and $A', B', C'$ be two triangles, that is, two triples of non-[[collinear|0AFNNTRTJE]] points.\nAssume $A \\ne A'$, $B \\ne B'$, $C \\ne C'$, and that the corresponding sides $AB$ and $A'B'$, $BC$ and $B'C'$, $CA$ and $C'A'$ are distinct lines.\nThen the lines $AA'$, $BB'$, $CC'$ pass through a common point if and only if the three points\n$$AB \\cap A'B', \\quad BC \\cap B'C', \\quad CA \\cap C'A'$$\nare collinear.\n\n## Remarks\n\nThe two triangles are then said to be in perspective from a point and in perspective from a line.\nFor two triangles in different planes of a [[projective space|ZT9A6JK8QF]] of dimension $3$, the proof is short: the three intersection points lie on both planes, hence on their common line.\nThe plane case follows by viewing the plane figure as a projection of a figure in space, or by a computation with [[homogeneous coordinates|FS9FRJ17D2]].\nThe converse direction is the dual statement, so it follows by the principle of [[duality|DCMP1A0138]].\n\nIn the Euclidean plane, if two triangles are homothetic, as $A' = 2A$, $B' = 2B$, $C' = 2C$ with centre the origin, the corresponding sides are parallel, so they meet on the line at infinity.\n\nThe theorem does not follow from the axioms of a projective plane alone.\nIt fails in the projective completion of the [[Moulton plane|3V9GYYXM1H]] and in other [[non-Desarguesian planes|HYE5FT4M34]].\nIt holds in a projective plane exactly when the plane is isomorphic to $\\mathbb{P}^2(D)$ for some division ring $D$.\n\n## Explanations\n\n### L1 human\n\nDraw two triangles so that the lines through matching corners all meet at one point, as if one triangle were the shadow of the other cast by a lamp.\nThen extend matching sides until they cross: the three crossing points always lie on one straight line.\nThe statement only talks about points and lines, with no lengths or angles.\n\n### L2 human\n\nDesargues's theorem is the dividing line between planes that come from algebra and planes that do not.\nA projective plane in which it holds can be coordinatized by a division ring.\nIt holds automatically in every projective space of dimension at least $3$, so the [[Veblen–Young theorem|NPNNXWMVCC]] needs no extra axiom there.\n[[Pappus's hexagon theorem|B6C3J2GJWP]] implies it, by Hessenberg's theorem, and Pappus holds exactly when the coordinate division ring is commutative.\nThe little Desargues theorem, where the centre lies on the axis, characterizes [[Moufang planes|7F4HSNJJH8]].\n",
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