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: The implication is then $$(R,PQ)$$—that point $R$ is incident with line $\overbar{PQ}$.
 * Points: $$\{A,B,C,a,b,c,P,Q,R,O\}$$
 * Lines: $$\{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}$$
 * Incidences (in addition to obvious ones such as $$(A,AB)$$): $$\{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}$$

Famous examples
Desargues' theorem holds in a projective plane $P$ if and only if $P$ is the projective plane over some division ring (skewfield) $D$ — $$P=\mathbb{P}_{2}D$$. 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 $$\mathbb{P}_{2}D$$ if and only if $D$ is a field; it corresponds to the identity $$\forall a,b\in D, \quad a\cdot b=b\cdot a$$.
 * Fano's axiom (which states a certain intersection does not happen) holds in $$\mathbb{P}_{2}D$$ if and only if $D$ has characteristic $$\neq 2$$; it corresponds to the identity $a + a = 0$.