It is well-known that Euclid's axioms for geometry are not up to modern standards of rigor: in particular, there are a lot of times when he used "obvious" facts about the geometric objects which were neither stated in, nor derivable from, the axioms.
For example, in his very first proposition, Book I Proposition I, he sets out to construct an equilateral triangle. Seems easy enough, right? He gives a construction involving two circles drawn at the ends of a line segment which is to be one side of the triangle and uses this to produce the remaining two sides. Trouble is, he has no axioms which guarantee that the circles will intersect, as they must to produce the third vertex of the triangle! Another, even more subtle, problem is that when you draw the segments to the third vertex, how do you know what you form is in fact a triangle? You need another axiom for that -- specifically, the problem is you don't know the sides don't meet somewhere else than the new point before they get there. This trouble was actually known even back in old Greece: Zeno of Sidon pointed this one out.
So that makes me wonder: just what can you prove from Euclid's axioms, definitions, etc. without resorting to any hidden, "obvious" assumptions? If you can't so much as even get something as basic as whether the points you are attempting to construct by crossing lines and circles really exist, it seems the answer is "not a whole heck of a lot!" Or am I wrong?
You can see :
It is a detailed analysis of Euclid's Elements logical structure. In particular, Ch.1.2 : Book I of the Elements , is devoted to the reconstruction of Book I, with graphical maps depicting the logical dependencies between Propositions.
You can use them, to check what Prop depends from the "flawed" ones : I mean those Prop of Book I which have some "hidden assumptions" asking for additional axioms not stated by Euclid.