In the Elements things that are equal can coincide perfectly with one another. However we cannot show things to be equal by "applying" one to another.
If two lines are equal, why can't we map one to the other and logically assume they share the same points afterwards, such as in the case of Book 1, Proposition 4 of The Elements
From what I've read, it seems the actual transformations required are not given as axioms which is the problem. But, ruler and compass transformations can map a line to any point on the plane, and its only left to assume that if two lines are equal, and share a point, that they must share the other point ( with some rotation. )
Read this statement from the link above: " If the triangle ABC is superposed on the triangle DEF, and if the point A is placed on the point D and the straight line AB on DE, then the point B also coincides with E, because AB equals DE."
The first part sounds much like it could be derived from I.2, but isn't. It seems the main problem from what I've read is that the required transformations do not have postulates, and that even if that doesn't matter. If two lines are congruent, it needs to be proved they may coincide and share points like Euclid describes.
Can someone verify this? Thank you!