In some lists of statements equivalent to the parallel postulate (such as Which statements are equivalent to the parallel postulate?), one can find the Pythagorean theorem. To prove this equivalence one has first to state the pythagorean theorem in neutral geometry (I name 'neutral geometry' a geometry in which parallel lines do exist but with the parallel postulate removed).
If one start with an axiom system like Birkhoff's postulates which assume reals numbers and ruler and protractor from the beginning then there is no problem stating the Pythagorean theorem.
My question is how one can one state the Pythagorean theorem in a neutral synthetic geometry based on axioms such as Hilbert's axioms group I II III or Tarski's axioms $A_1-A_9$ ?
It is possible to define segment length in neutral Tarski's or Hilbert's geometries as an equivalence class using the congruence ($\equiv$) relation. It is also possible to define the congruence of triangles.
However, the geometric definition of multiplication as given by Hilbert assume the parallel postulate. The existence of a square is equivalent to the parallel postulate.