Herleitung von Skalarprodukten aus Symmetrieprinzipien
  This is an attempt to model ambient space as a three-dimensional real affine space with a distinguished group of automorphisms containing the translations and acting freely and transitively on pairs consisting of a half-plane together with a half-line on its boundary. From there the existence of an invariant scalar product is deduced, which then also implies Pythagoras theorem in a quite precise form. This is in contrast to the usual procedure to model ambient space by asking for a distinguished scalar product and using Pythagoras theorem as known from high school to connect with reality. 
Wolfgang Soergel