NJ Wildberger says in this video that there's a logical flaw in Euclid's construction of the triangle, that you're not really able to know (apart from the picture) if the circles intersect. He also provided no further comments about that. So, how's that?
As the speaker says: How do we know that the circles intersect? The axiom system in Euklid's Elements is in fact not strong enough to show that these circles intersect. Essentially, there is missing a notion of "between" and completeness, which has been corrected later by Hilbert. To see the problem immediately, note that you can do Euklidean geometry in $\mathbb Q^2$.