Let the usual five postulates of Euclid been given.
Let's take also this postulate: "If two points lies on the same plane, the whole straight line joining the two points lies on that plane".
Is it possible to prove that if two straight lines intersect each other, there exist a plane containing them both?
EDIT: I don't want to prove ULTRA-RIGOROUSLY in Hilbert-style (for example I don't question the existence of planes and I don't bother about uniqueness). I want to prove it as Euclid would prove it.
With this further postulate it's simple to prove these two facts:
If Two planes that meets in a point, meets each other in a straight line
Proof: Let $\alpha$ and $\beta$ the two given planes meeting in A. Take any points B and C on the plane $\alpha$, and not on the plane $\beta$ but on the same side of it. Join AB, AC and produce BA to F (on the opposite side of $\beta$). Join CF. Then since B and F are on opposite sides of the plane $\beta$, C and F are also on opposite sides of it. Therefore CF must meet the plane $\beta$ in a point G. Then since A, G are both in each of the planes $\alpha$ and $\beta$, the straight line AG is in both planes.
If two straight lines AB and AC meeting in A lies on the same plane, then the triangle ABC lies on the same plane.
EDIT 2: You proposed me counterexamples in 4 dimensions, solutions with algeabric geometry... etc. I repeated 1000 times that I would like to find a solution suitable in 300 BC!!! It's not to hard to understand that Galois Theory or Riemann surfaces cannot be used! xD
Here it is an example of solution that I DON'T LIKE by Robert Simson
Let any plane pass through the straight line AB (produced indefinitely) as axis until it passes through the point C. Then, since the points A, C are in this plane, the straight line AC is on the plane and so it is BC.
I don't like the using of rotating motion, but this is in fact a solution. I want to find alternative proofs.