Theorems in projective geometry | Incidence geometry

Intersection theorem

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 A and B (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 A and B 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 β€”that point R is incident with line PQ. (Wikipedia).

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Related pages

Desarguesian plane | Shimshon Amitsur | Fano plane | If and only if | Characteristic (algebra) | Projective geometry | Field (mathematics) | Division ring | Tensor product | Theorem | Incidence structure | Module (mathematics) | Pappus's hexagon theorem