On the Proof of the Fifth Postulate (the Parallel Axiom) in Euclidean Geometry

Abstract

The framework of Euclidean geometry lacks a definitive criterion to distinguish straight lines from curves, prompting the introduction of the Fifth Axiom (the Parallel Postulate) as a logical patch. However, this postulate can be proven without new axioms by utilizing the concept of area. The core distinction lies in the fact that equal line segments can coincide after translation on a plane, but not necessarily on curved surfaces. This translational property enables the proof of the Parallel Postulate on a plane without contradicting the existence of non-Euclidean geometries.