You were done before you answered it..

One more on the more obvious side, shouldn't a line still be, albeit a degenerate, case of a Euclidean space on which one can do Euclidean geometry? That's why I didn't have A1

There are a few more subtle things that are effectively different.

@moishe kohan I don't see anything missing from your reinterpretation except from the A3. Isn't this supposed to be a version of the playfairs axiom? I mean, and maybe someone said this, my question could (almost) be stated as "is every geodesic complete metric space with playfairs axiom an euclidian space"?

I will take note on the substantially of the comments, but it wasn't and isn't my intention really to dismiss anyone or go into arguments.

I did point out every time I patched or updated something and asked if it could be pardoned in the interest of the question. I still think it wasn't changed much..

I only said to moishe kohan that saying this question is not research level and than answering it by a research level text reference seems contradictory.

I mean how would you approach a this, if I see a detail that needs to be polished what do I do?

I could have expanded on The angle measure, but it would be In vain since as you said the question is dismissed.

I appreciate each comment, no matter how parts of some even yours seem more than silly.

I don't get the toxicity.. You give up cause you're wrong, I hate to say it but it's true.. Why are you being so dismissive...

You said geodesics here have only 2 points, that's what the continuity axiom negates and that's what you said is wrong with 3 by itself, that that it allows non unique geodesics..

Even originally I had one additional axiom for continuity, as I posted I realised it could have been stated in a better way but then there were already comments and I didn't want to change the question too much.

That counter example didn't work even with the original formulation.

But look at the edit.

But that's OK, it's a good answer attempt I'm not sure yet if it's correct.

So why I said you answer is to involved for math SE is because you require i know what an Lp space is and then refer to a graduate to research level paper. So that by definition then is an answer on that level so how then is the question not provided the answer is suitable for it?