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I have studied what Heath, Russell, and what others have said about the proposition 4 of the Book I of the Elements. So far, I understand the "problems" they see with using superposition, but I still can't wrap my head around the matter of I could use coinciding figures to make the proof.

I think that I don't have to worry about the motion of the triangles in the plane, or the fact that I still don't have anyway to measure angles. I think that the problem could be seen like that:

Suppose that I have a triangle ABC, and that I have the figure consisting of the segment DE and the segment EF, such as the segment DE coincides with the segment AB and the segment EF coincides with the segment BC (thus A=D, B=E and C=F). The only way that I can make a segment connecting the poings D and F is if the resulting segment DF coincides with the segment AC. Since such segment would make a triangle DEF, that triangle would coincide with the triangle ABC and from that I can say that both triangles are equal.

What is the problem in my rationale? I started with a figure consisting of 2 segments spread by an angle (I don't have to worry about measures or "equal angles" because I'm using coinciding points to ABC as source for the figure) and from that I proved that I can make only one triangle out of those components, and this triangle in equal to ABC.

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I would think this falls under Common Notion $4$ ("Things which coincide with one another equal one another." This Common Notion may have been added much later, by someone who noticed the possible gap. – André Nicolas Mar 26 '12 at 22:55
I'm no expert, but I think CM4 is also from the same time as the other propositions, mostly because the common notions and postulates are needed for the propositions to work and are mentioned in the some other propositions. The CM4 in invoked in this proposition, but here we are talking of two triangles as being congruent/equal (and not coinciding), I used the CM4 even more to construct my rationale coinciding the points and from there extend to any figure in a similar pattern. – Luiz Borges Mar 26 '12 at 23:26

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