Problems, procedures and indespensability of superposition in the proofs of congruence of triangles in I.4 and I.8 of Euclid's Elements

Abstract

The Book I of Euclides’s Elements begins with three propositions that ask the solution to three problems. Unlike what happens in other propositions, they do not constitute an assertion of properties or relations between geometric objects. Instead, they request actions whose execution generates geometric objects. The propositions that demand the obtaining of a geometric object are typically presented as problems. Their resolution is normally accompanied by the solution of a second problem: the problem of demonstrating that the previously offered solution is correct. Only from the fourth proposition onwards will the theorems of the Elements begin to appear. Next, we will consider, in a preliminary manner, the implications of adopting an approach to geometry in which propositions are interpreted as problems, including those read as theorems, and also that the solution to a construction problem is a procedure or algorithm based on the postulates. Finally, within the outlined framework, we will offer alternative proofs for the first two triangle congruence theorems without any appeal to the famous operation of superposition.