In Book I Proposition 10 of the Elements, Euclid performs the bisection (i.e. finding a midpoint) of a line segment. In the course of doing so, he uses Book I Proposition 4, the Side-Angle-Side Theorem, which proved with his controversial method of superposition. My question is, could Euclid have proven Book I Proposition 10 without relying on Book I Proposition 4, so that the result wouldn't depend on the method of superposition?
The reason I ask is that I think I've found a simpler proof of Euclid's Book I Proposition 2 (involving the transferring of distances), but it relies on bisection, so I'm hoping that I'm not implicitly using the method of superposition.
Any help would be greatly appreciated.
Thank You in Advance.
The unavoidable step in the proof is the principle that opposite sides of a parallelogram are equal.
In Euclidean geometry this is proved using the congruence axiom that you want to avoid, but really it is a weaker principle from the subsystem of affine geometry that does not include the concepts of angle or rotation. In effect it is the Euclidean axiom restricted to superpositions that are parallel translations. (Maybe 180 degree rotations could be implicitly needed for some arguments, but they are avoidable for the midpoint proof by using a chain of several parallelograms, and they might be avoidable in general).
The two subsegments on either side of the midpoint are different geometric objects, and without some comparison principle no relation between them can be proved. There would be one segment divided into two parts but no other facts derivable. The minimal sufficient principle is the one about parallelograms.