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In mathematics, Pappus' hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points ''A'', ''B'', ''C'', and another set of collinear points ''a'', ''b'', ''c'', then the intersection points ''X'', ''Y'', ''Z'' of line pairs ''Ab'' and ''aB'', ''Ac'' and ''aC'', ''Bc'' and ''bC'' are collinear, lying on the ''Pappus line''. These three points are the points of intersection of the "opposite" sides of the hexagon ''AbCaBc''. It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring.〔Coxeter, pp. 236-7〕 Projective planes in which the "theorem" is valid are called pappian planes. The dual of this incidence theorem states that given one set of concurrent lines ''A'', ''B'', ''C'', and another set of concurrent lines ''a'', ''b'', ''c'', then the lines ''x'', ''y'', ''z'' defined by pairs of points resulting from pairs of intersections ''A''∩''b'' and ''a''∩''B'', ''A''∩''c'' and ''a''∩''C'', ''B''∩''c'' and ''b''∩''C'' are concurrent. (''Concurrent'' means that the lines pass through one point.) Pappus' theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines. Pascal's theorem is in turn a special case of Cayley–Bacharach_theorem. The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus' theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of ''ABC'' and ''abc''.〔However, this does occur when ''ABC'' and ''abc'' are in perspective, that is, ''Aa'', ''Bb'' and ''Cc'' are concurrent.〕 This configuration is self dual. Since, in particular, the lines ''Bc'', ''bC'', ''XY'' have the properties of the lines ''x'', ''y'', ''z'' of the dual theorem, and collinearity of ''X'', ''Y'', ''Z'' is equivalent to concurrence of ''Bc'', ''bC'', ''XY'', the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges. ==Proof== Choose projective coordinates with :, , , . On the lines ''AC'', ''Ac'', ''AX'', given by , , , take the points ''B'', ''Y'', ''b'' to be :, , for some ''p'', ''q'', ''r''. The three lines ''XB'', ''CY'', ''cb'' are , , , so they pass through the same point ''a'' if and only if . The condition for the three lines ''Cb'', ''cB'' and ''XY'' , , to pass through the same point ''Z'' is . So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so . Equivalently, ''X'', ''Y'', ''Z'' are collinear. The proof above also shows that for Pappus' theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus' theorem implies Desargues' theorem.〔According to , Hessenberg's original proof is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by .〕 In general, Pappus' theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus' theorem does not hold are Desarguesian projective planes over noncommutative division rings, and non-Desarguesian planes. The proof is invalid if ''C'', ''c'', ''X'' happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pappus's hexagon theorem」の詳細全文を読む スポンサード リンク
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