In a Euclidean right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs: if the legs are a and b and the hypotenuse c, then c^2 = a^2 + b^2.

Principle

The theorem encodes the metric relation induced by orthogonality: orthogonal components contribute additively to squared distance, reflecting the Euclidean inner product structure.

Demonstration

Classic numerical example: a 3–4–5 right triangle satisfies 5^2 = 3^2 + 4^2 because 25 = 9 + 16. Proofs are manifold (geometric rearrangement, similar triangles, algebraic).

Misapplication

Applying the identity c^2 = a^2 + b^2 to a non‑right triangle without the correction term of the law of cosines, or using it in non‑Euclidean metrics such as taxicab geometry where the squared‑distance relation fails.

Consequence

It provides a direct method to compute distances, check for right angles from side lengths, and underpins coordinate geometry, orthogonal projections, and the notion of Euclidean norm and orthonormal bases.

Reversal

The Law of Cosines generalizes the theorem: for non‑right triangles c^2 = a^2 + b^2 − 2ab cos(γ); conversely, if c^2 < a^2 + b^2 the triangle is acute, if greater it is obtuse.

Boundary

Valid in Euclidean (flat) geometry with the standard notion of distance; not valid in curved spaces, in discrete metrics, or without an inner‑product structure that yields squared‑distance additivity.

Semantic Tension

Competes with generalized distance concepts (normed spaces with other p‑norms) and with discrete path metrics; the tension is between Euclidean squared additivity and alternative measures of length.

Synthesis

The Pythagorean theorem is the fundamental Euclidean identity relating orthogonality and squared distances in right triangles; it is a special case of the law of cosines and a cornerstone for metric, algebraic and geometric constructions.