Q:

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.

A:

No. 
Euclid made a number of unstated assumptions, which the German mathematician David Hilbert recognized.  Hilbert addressed them in his text Grundlagen der Geometrie (Foundations of Geometry) by adding new axioms.  Among the omissions made by Euclid was the concept of what it means for an object to be inside a closed curve, a question answered by the French mathematician Camille Jordan, a result known as the Jordan curve theorem.

