##  [Pythagorean Theorem](/pythagorean-theorem-0) 

 Definition

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

Principle

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

 

 

 

 

 





## Demonstration

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

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

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

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

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

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

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.