Pappos Theorem  --- ten proofs and three variations;
OR: How to read in equations

J"urgen Richter-Gebert (ETHZ)

The main topic of this talk is to show how inspection of 
algebraic equations can be used to generalize proofs of 
geometric theorems. We take the well known Theorem of 
Pappos about nine points and nine lines in the projective plane
as a starting point for our investigation. Starting from there
we will look at several different ways of proving this theorem.
Each way of proving leads to different kinds of generalization.
By this we will obtain a large collection of geometric incidence 
theorems including many well known theorems in euclidean and 
projective geometry. In particular we will see, that the structure
of cycles in geometric simplicial complexes is responsible
for the algebraic cancelation patterns that arise in many proofs
of geometric theorems.