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 and . The "theorem" states that, whenever a set of objects satisfies the incidences, 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:
- Points:
- Lines:
- Incidences :
The implication is then —that point is incident with line.holds in a projective plane if and only if is the projective plane over some division ring (skewfield
