I have thought about the question for some time and now I decide to ask community, I belive some good answers exist.
Consider two facts about Euclidean plane:
- Suppose $c$ be a circle with radius $OA$ and $B$ be such point of plane, that $OB$ is greater than $OA$, then there is a point $C$ of intersection.
- Suppose $c_{1}$, $c_{2}$ be two circles with radii $O_{1}A_{1}$ and $O_{2}A_{2}$, such that sum of $O_{1}A_{1}$ and $O_{2}A_{2}$ is greater than $O_{1}O_{2}$ and difference of $O_{1}A_{1}$ and $O_{2}A_{2}$ is less than $O_{1}O_{2}$, then there exists a point of intersection of $c_{1}$ and $c_{2}$.
As I see Euclid in first book does not give any justification of the facts. What would be the reason, a cause of such gap, and what are possibilities to fill the gap? It might be the change of postulates about plane, or additional proposition / propositions in flow of the first book. What is known about that?
You don't need completeness. Historically the first mathematician who fixed this was Pasch; see Pasch axiom which is far more elementary than completeness and holds in many fields where completeness fails.
You might want to consult Greenberg's book Euclidean and non-Euclidean geometry for the background.