Pythagorean theorem

Mathematical statement

For every right triangle in the Euclidean plane, if the legs have lengths a and b and the hypotenuse has length c, then a squared plus b squared equals c squared.

Representative formula (exact TeX)

a^2+b^2=c^2

Assumptions

• The geometric setting is the Euclidean plane.
• The triangle has a right angle.
• The symbols a and b denote the leg lengths, and c denotes the hypotenuse length.
• The side lengths are nonnegative real numbers.

Symbol scope

• a — first leg length
• b — second leg length
• c — hypotenuse length

Intuition, definition, and misconceptions

The theorem does not hold for every triangle; it applies to right triangles. The hypotenuse is the longest side and lies opposite the right angle.

The squared relation can be viewed as an area identity: the combined areas of squares built on the two legs equal the area of the square built on the hypotenuse.

A program can check specific numerical triples or search for integer Pythagorean triples, but finite checking cannot replace a universal proof of the theorem.

Lean and Mathlib first prove a vector-angle form. IsRightTriangleSideModel then explicitly binds a=‖x‖, b=‖y‖, and c=‖x+y‖ to derive the equation a²+b²=c² used by the encyclopedia entry.

A common mistake is to apply a²+b²=c² to an arbitrary triangle, or to treat the formalized vector side-length model as though equivalence with every affine representation had already been proved automatically.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Right triangle — The theorem applies to this geometric object.
• Euclidean length — The squared relation depends on Euclidean distance.

Verification boundary

Python and FELRA provide finite computational evidence. Lean and Mathlib prove both the vector core and OCME's declared side-length binding model. Equivalence from an arbitrary affine triangle representation to this model still requires a separate proof.

formalized_declared_side_model

• evidence-sha256-49083d0529f2b530b946fd6f30a313b0341a8989bf0f6de9adcdeb398a2b4466
• evidence-sha256-97ad4a4b529de8c77662f823ad0966c24cfe1c121af8cbb4320135b9ea849448
• evidence-sha256-4d31c152012f8b561749e354f3789344c91e1fea26cead1468e95417b2e84752
• evidence-sha256-701b8066d363dadf2cc1ec21bc0d21f7c680a6eb187a37fd309c0f2e4c86d08e

Next obligation

Prove equivalence between arbitrary Euclidean affine-triangle vertex representations and IsRightTriangleSideModel, including vertex ordering, edge-vector direction, and preservation of distance.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.