A single instance of the Pythagorean theorem implies the parallel postulate

Abstract

This note could find use as enrichment material in a course on the classical geometries; its preliminary results could also be used in an advanced calculus course. It is proved that if a, b and c are positive real numbers such that a ₂ + b ² = a ², then cosh (a) cosh (b) > cosh (c). The proof of this result uses an inequality involving combinatorial symbols and properties of absolutely convergent infinite series. As a corollary, it follows that if a neutral geometry G contains at least one right triangle which satisfies the conclusion of the Pythagorean Theorem, then G is (isomorphic to) Euclidean geometry and, hence, satisfies Playfair's form of the parallel postulate.