Proof of Playfair’s Axiom Hits a Roadblock

Authors: Prashanth R. Rao

The Playfair’s axiom is considered an equivalent of Euclid’s fifth postulate or parallel postulate in Euclidean planar geometry. It states that in a given plane, with a line in the plane and a point outside the line that is also in the same plane, one and only one line passes through that point that is also parallel to the given line. Previous proofs of Euclid’s postulate or the Playfair’s axiom have unintentionally assumed parallel postulate to prove it. Also, these axioms have different results in hyperbolic and spherical geometries. We offer proof for the Playfair’s axiom for subset of cases in the context of plane Euclidean geometry and describe another subset of cases that cannot be proven by the same approach.

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[v1] 2018-03-16 18:19:17

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