Abstract
On the basis of the theoryε − 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 toε −. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.
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Pambuccian, V. Splitting the Pasch axiom. J Geom 56, 126–130 (1996). https://doi.org/10.1007/BF01222689
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DOI: https://doi.org/10.1007/BF01222689