In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure â consisting of points, lines, and possibly higher-dimensional objects and their incidences â 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:
The implication is then âÂÂthat point is incident with line .
Desargues' theorem holds in a projective plane if and only if is the projective plane over some division ring (skewfield) â . 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.