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).
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
Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring (skewfield) D —
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 P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.