# 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 —that point *R* is incident with line *PQ*.

## 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* — . 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 if and only if
*D*is a field; it corresponds to the identity . - Fano's axiom (which states a certain intersection does
*not*happen) holds in if and only if*D*has characteristic ; it corresponds to the identity*a*+*a*= 0.

## References

- Rowen, Louis Halle, ed. (1980).
*Polynomial Identities in Ring Theory*. Pure and Applied Mathematics.**84**. Academic Press. doi:10.1016/s0079-8169(08)x6032-5. ISBN 9780125998505. - Amitsur, S. A. (1966). "Rational Identities and Applications to Algebra and Geometry".
*Journal of Algebra*.**3**(3): 304–359. doi:10.1016/0021-8693(66)90004-4.