Euclidean geometry and its fifth axiom are interesting, but unrelated to the Banach-Tarski paradox. I don't get the point you're trying to make. It's not a "wider context", it's a different thing altogether.
Also, as someone pointed out in the linked thread, you're completely glossing over the theory of measurable sets.
Also, as someone pointed out in the linked thread, you're completely glossing over the theory of measurable sets.