You're confusing arbitrary with random. We have chosen the axioms of Euclidean geometry because they are useful, but we could very easily chose different axioms

You're confusing arbitrary with random. We have chosen the axioms of Euclidean geometry because they are useful, but we could very easily chose different axioms and derive a different geometric system. We didn't choose the Euclidean axioms randomly, but they weren't an a priori given either.