On the equivalence of Playfair's axiom to the parallel postulate

2019/03/12 by Elizabeth T. Brown, Emily Castner, Brown, Elizabeth T. +9
Mathematics · #51F05 #51M05 #FOS: Mathematics #History and Overview (math.HO) #Metric Geometry (math.MG) #math.HO #math.MG #msc:51F05 #msc:51M05

paper · pdf · doi:10.48550/arxiv.1903.05233

5 pages, 3 figures

arxiv created 2019/03/12 · arxiv updated 2019/03/15

Abstract

We show that the classical equivalence of Euclid's parallel postulate and Playfair's axiom collapses in the absence of triangle congruence. In particular, we construct a non-SAS geometry that models the Playfair axiom but not the parallel postulate.

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