Relation To Pappus' Theorem
Pappus's hexagon theorem states that, if a hexagon AbCaBc is drawn in such a way that vertices a, b, and c lie on a line and vertices A, B, and C lie on a second line, then each two opposite sides of the hexagon lie on two lines that meet in a point and the three points constructed in this way are collinear. A plane in which Pappus's theorem is universally true is called Pappian. Hessenberg (1905) showed that Desargues's theorem can be deduced from three applications of Pappus's theorem (Coxeter 1969, 14.3).
The converse of this result is not true, that is, not all Desarguesian planes are Pappian. Satisfying Pappus's theorem universally is equivalent to having the underlying coordinate system be commutative. A plane defined over a non-commutative division ring (a division ring that is not a field) would therefore be Desarguesian but not Pappian. However, due to Wedderburn's theorem, which states that all finite division rings are fields, all finite Desarguesian planes are Pappian. There is no known, satisfactory geometric proof of this fact.
Read more about this topic: Desargues' Theorem
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