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Desargues's theorem
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{{short description|Two triangles are in perspective axially if and only if they are in perspective centrally}} [[Image:Desargues theorem alt.svg|thumb|350px|Perspective triangles. Corresponding sides of the triangles, when extended, meet at points on a line called the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues's theorem states that the truth of the first condition is [[necessary and sufficient]] for the truth of the second.]] In [[projective geometry]], '''Desargues's theorem''', named after [[Girard Desargues]], states: :Two [[triangle]]s are in [[perspective (geometry)|perspective]] ''axially'' [[if and only if]] they are in perspective ''centrally''. Denote the three [[vertex (geometry)|vertices]] of one triangle by {{math|''a'', ''b''}} and {{math|''c''}}, and those of the other by {{math|''A'', ''B''}} and {{math|''C''}}. ''Axial [[perspectivity]]'' means that lines {{math|{{overline|''ab''}}}} and {{math|{{overline|''AB''}}}} meet in a point, lines {{math|{{overline|''ac''}}}} and {{math|{{overline|''AC''}}}} meet in a second point, and lines {{math|{{overline|''bc''}}}} and {{math|{{overline|''BC''}}}} meet in a third point, and that these three points all lie on a common line called the ''axis of perspectivity''. ''Central perspectivity'' means that the three lines {{math|{{overline|''Aa''}}, {{overline|''Bb''}}}} and {{math|{{overline|''Cc''}}}} are concurrent, at a point called the ''center of perspectivity''. This [[intersection theorem]] is true in the usual [[Euclidean plane]] but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following [[Jean-Victor Poncelet]]. This results in a [[projective plane]]. Desargues's theorem is true for the [[real projective plane]] and for any projective space defined arithmetically from a [[field (mathematics)|field]] or [[division ring]]; that includes any projective space of dimension greater than two or in which [[Pappus's hexagon theorem|Pappus's theorem]] holds. However, there are many "[[non-Desarguesian plane]]s", in which Desargues's theorem is false.
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