Helge on Nostr: I think e and pi being properly transcendental implies that one cannot make a right ...

I think e and pi being properly transcendental implies that one cannot make a right triangle out of them. I'm too out of practice to sketch an argument. Probably, one needs to invoke Euler's equation somewhere. So not being a right triangle has a reason.
For the likely hood to be close to a right triangle, one probably needs to run some estimates. It clearly depends on what exponents you allow. Using 2.5 is already quite the edge case.