On the basis of the theory ε- of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: "In any right triangle the hypotenuse is greater than the leg" and "If ∠AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD." This represents an attempt to split the Pasch axiom with respect to ε-. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.
ASJC Scopus subject areas
- Geometry and Topology