  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.
