Pythagorean Theorem

The Pythagorean Theorem is truly seminal. Its particular proof offered reveals linkages amongst quantification in Geometry; the concept of parallelism; similar triangles and the self-similarity property of right-triangles; Pasch’s Axiom; and Eudoxan theory of proportions. Besides refuting the seemingly self-evident commensurability assumption, the Theorem offers a perceptive geometric approach to area measure of plane figures. It forms the bedrock of Trigonometry. It is equivalent to Brahmagupta’s Theorem on cyclic quadrilateral, and to Heron’s area formula for a triangle. It furnishes an arithmetico-algebraic basis for Coordinate Geometry, inspiring notions of multi-dimensional spaces and metric spaces. By its distinctive form, the Theorem has generated various conjectures and concepts, linked to Fermat’s Last Theorem, congruent numbers, and non-Euclidean geometries.