# Congruence of nondegenerate triangles with explicit correspondence

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

In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection.

mko-congruence · definition · v0.10.0



## Mathematical statement

In one equal-unit orthonormal Euclidean plane, let A=(A1,A2,A3) and B=(B1,B2,B3) be nondegenerate triangles with explicit correspondence Ai↔Bi. This pulse uses the SSS congruence slice: all three corresponding side lengths agree. Compare their squares exactly; reflection reverses orientation without defeating congruence.





## Representative formula (exact TeX)

G(A,B)





## Assumptions


- Each triple contains distinct noncollinear points. Except in triangle-eligibility questions, nondegeneracy is established before applying this entry.
- Coordinates in the exercises are safe integers, and squared sides are ordered 12,13,23. Do not guess a label permutation or judge by visual appearance.
- Eligibility questions test noncollinearity through the nonzero 2×2 determinant of (A2−A1,A3−A1), without inheriting an earlier area or Pythagorean proof.





## Symbol scope


- G — scoped labeled-triangle congruence relation
- A — first labeled nondegenerate triangle
- B — second triangle with explicit correspondence





## Intuition, definition, and misconceptions

Start with the explicitly labeled SSS decision, not a possible rearrangement of an unlabeled figure. Opposite orientations are allowed; a sign change in oriented data does not disprove reflected congruence.

Any unequal corresponding squared side rejects this correspondence. Select the first mismatch in the fixed order 12,13,23; if all agree, report none. Rejecting this correspondence does not reject every other vertex permutation.

A corresponding-side question first confirms full congruence, then reads a specified permitted side. An irrelevant third point or separate triangle translation does not create a new length computation. Eligibility questions separately admit collinear or repeated points as invalid candidates.

Common errors include ignoring nondegeneracy or labels, guessing congruence from a sketch, or rejecting it because reflection reverses orientation.





## Adopted derivation guide

[Congruence: deciding congruence of labeled triangles by the SSS slice](https://ocme.evemisslab.com/en/derivations/derivation-mko-congruence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Triangle](https://ocme.evemisslab.com/en/mko/mko-triangle/) — First establish three distinct noncollinear points.
- [Euclidean distance in an equal-unit orthonormal frame](https://ocme.evemisslab.com/en/mko/mko-distance-formula/) — Compare corresponding side lengths through their squares.
- [Determinants of real 2×2 matrices](https://ocme.evemisslab.com/en/mko/mko-determinant/) — Use the exact coordinate criterion for noncollinearity.





## Verification boundary

Labeled integer-coordinate SSS instances can be recomputed; no new formal bridge for general isometries, congruence, or SSS is attached.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bridge EuclideanSpace noncollinearity, corresponding-distance equality, and triangle SSS while retaining reflections and explicit labels.





## 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/congruence.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-congruence/index.md)