# Pythagorean theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

In a right triangle, the sum of the squares of the two leg lengths equals the square of the hypotenuse length.

mko-euclid-pythagorean-theorem · theorem · v0.8.0



## 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

[Pythagorean theorem: a proof by dot products in the coordinate model](https://ocme.evemisslab.com/en/derivations/derivation-mko-euclid-pythagorean-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Right triangle](https://ocme.evemisslab.com/en/mko/mko-right-triangle/) — The theorem applies to this geometric object.
- [Euclidean length](https://ocme.evemisslab.com/en/mko/mko-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](https://ocme.evemisslab.com/data/evidence/49083d0529f2b530b946fd6f30a313b0341a8989bf0f6de9adcdeb398a2b4466.json)
- [evidence-sha256-97ad4a4b529de8c77662f823ad0966c24cfe1c121af8cbb4320135b9ea849448](https://ocme.evemisslab.com/data/evidence/97ad4a4b529de8c77662f823ad0966c24cfe1c121af8cbb4320135b9ea849448.json)
- [evidence-sha256-4d31c152012f8b561749e354f3789344c91e1fea26cead1468e95417b2e84752](https://ocme.evemisslab.com/data/evidence/4d31c152012f8b561749e354f3789344c91e1fea26cead1468e95417b2e84752.json)
- [evidence-sha256-701b8066d363dadf2cc1ec21bc0d21f7c680a6eb187a37fd309c0f2e4c86d08e](https://ocme.evemisslab.com/data/evidence/701b8066d363dadf2cc1ec21bc0d21f7c680a6eb187a37fd309c0f2e4c86d08e.json)





## 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/pythagorean-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-euclid-pythagorean-theorem/index.md)