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: { A , B , C , a , b , c , P , Q , R , O } Lines: { A B , A C , B C , a b , a c , b c , A a , B b , C c , P Q } Incidences (in addition to obvious ones such as ( A , A B ) ): { ( O , A a ) , ( O , B b ) , ( O , C c ) , ( P , B C ) , ( P , b c ) , ( Q , A C ) , ( Q , a c ) , ( R , A B ) , ( R , a b ) } The implication is then ( R , P Q ) —that point R is incident with line PQ.
Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring (skewfield} D — P = P 2 D . 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 P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.
Pappus's hexagon theorem holds in a desarguesian projective plane P 2 D if and only if D is a field; it corresponds to the identity ∀ a , b ∈ D , a ⋅ b = b ⋅ a .Fano's axiom (which states a certain intersection does not happen) holds in P 2 D if and only if D has characteristic ≠ 2 ; it corresponds to the identity a + a = 0.