In projective geometry, an intersection theorem or incidence theorem is an incidence structure consisting of points, lines, and possibly higher-dimensional objects and their incidences, together with a pair of nonincident 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 matched up with the objects of the incidence structure in a way that preserves incidence), then the objects corresponding to A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is rather a property which some geometries satisfy but not others.
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.
Read more about Intersection Theorem: Famous Examples
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