Is possible to prove any theorem with Euclidean axioms?

In 2D geometry, where the five axioms of Euclid are right, can I prove any geometric theorem? like pythagorean theorem, bisector theorem, law of parallelogram, etc.?

I think that it could, because the axioms are the basis of all this geometry and therefore, more complex theorems should be able to express themselves in terms of their "elementary particles" (analogy). But this is just an intuition.

• I don’t think you are supposed to call something a theorem if nobody can prove it, so, tautologically, yes, theorems have proofs, if that’s what you mean. Whether or not the random assertions you have in mind are probable from the random axioms you have in mind is a completely different issue , one which we have insufficient data on to answer. – rschwieb Sep 24 '18 at 10:29
• I think you are asking if every conceivable true statement about geometry can be proven from Euclid's axioms. The term for that is completeness - you want to know if Euclid's axioms are complete. – Jack M Sep 24 '18 at 12:08
• To answer the question, I suspect the answer is no, by Godel's incompleteness theorem, because surely any reasonable formalization of Euclid's axioms would be able to talk about arithmetic (since Euclid's work involves notions of length and area). – Jack M Sep 24 '18 at 12:15