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In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects and (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (''i.e.'' can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects and must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't. For example, Desargues' theorem can be stated using the following incidence structure: *Points: *Lines: *Incidences (in addition to obvious ones such as ): The implication is then —that point is incident with line . == Famous examples == Desargues' theorem holds in a projective plane if and only if is the projective plane over some division ring (skewfield} — . The projective plane is then called ''desarguesian''. A theorem of Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane satisfies the intersection theorem if and only if the division ring satisfies the rational identity. *Pappus's hexagon theorem holds in a desarguesian projective plane if and only if is a field; it corresponds to the identity . *Fano's axiom (which states a certain intersection does ''not'' happen) holds in if and only if has characteristic ; it corresponds to the identity . 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「intersection theorem」の詳細全文を読む スポンサード リンク
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