[CaptainNegative]

This proof has almost nothing to do with Euclid. The Pythagoreans knew about it more than a century before his birth (Hippasus was apocryphally killed for divulging this proof), and the proof is widely believed to have only been inserted into Elements by others after Euclid's death.

[chx]

Yeah, this can't be done in the geometrical way Euclid worked

Mostly because you need the Archimedean property for it which can not be derived from Euclid's axioms.