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I searched a lot in the internet and also found several proofs, but I found some interdependent argumentation, meaning that for proving SAS one used ASA etc. Where can I find some elementary proofs of those theorems that don't require any of the other?

Hilberto1
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  • It's only circular if you use $A$ to deduce $B$ and then use $B$ to deduce $A$. – lulu Mar 25 '21 at 13:25
  • @Lulu thanks, I used the wrong word – Hilberto1 Mar 25 '21 at 13:27
  • Robin Hartshorne, Geometry: Euclid and Beyond – Mauro ALLEGRANZA Mar 25 '21 at 13:31
  • More broadly, it is difficult to answer your question without knowing what your concern is. You could use Euclid, say, though modern readers tend to feel that the use of axioms there is a little "flexible". Modern treatments often use one of those theorems as an axiom, from which the others are deduced. – lulu Mar 25 '21 at 13:31
  • You can actually proof all three are equivalent and the use anyone to show that all the dimensions of both the 2 triangles are equal – YOu will not know Mar 25 '21 at 13:32
  • @MauroALLEGRANZA thanks! – Hilberto1 Mar 25 '21 at 13:33
  • @MauroALLEGRANZA That text specifically uses $SAS$ as an axiom, for example. – lulu Mar 25 '21 at 13:33
  • @MauroALLEGRANZA but that's actually what I want to avoid, seeing it as an axiom... – Hilberto1 Mar 25 '21 at 13:35
  • @Lulu so there doesn't exist a fully rigorous proof that doesn't take it as an axiom? – Hilberto1 Mar 25 '21 at 13:36
  • @Descrates But, then, you need to specify which axiom system you prefer. Hartshorne is just following Hilbert...SAS is Hilbert's axiom $C6$. If you don't want that system, which one do you want? – lulu Mar 25 '21 at 13:37
  • "rigorous" only makes sense in the context of an axiom system. I think it is broadly agreed that Euclid's system isn't quite strong enough. I believe, but it;'s been years (decades, really) since I've thought about it, that some version of Superposition can be made into a workable axiom (but, as I say, I am not certain of this). – lulu Mar 25 '21 at 13:38
  • See Tarski's axioms for Geometry: the conguence theorems follows from Five-segments axiom. – Mauro ALLEGRANZA Mar 25 '21 at 13:41
  • Some sources – Mauro ALLEGRANZA Mar 25 '21 at 13:42
  • To be precise: Euclid uses Superposition in his proof. That's the notion that you can "pick up" an angle and move it somewhere else. Nothing in the axioms he provides allows for this. And, he knew this...there are many instances in the Elements in which Superposition would radically simplify the argument, but he refrains. – lulu Mar 25 '21 at 13:42

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