Pappus’ theorem and Desargues’ theorem

Abstract In Chapter 2 we saw that any field K gives rise to a projective plane PG(2, K). One very natural question to ask is: do all projective planes arise from a field in this way? In the present chapter we show that the answer is no. We do this by first proving that a classical theorem of Euclidean geometry, namely Pappus‘ theorem, holds in the projective plane PG(2, K) for all fields K, and then by exhibiting a projective plane in which Pappus‘ theorem does not hold. Another theorem of Euclidean geometry which may or may not hold in a given projective plane is Desargues‘ theorem. A famous result of projective geometry states that in any finite projective plane Pappus‘ theorem is implied by Desargues‘ theorem.