Congruence of nondegenerate triangles with explicit correspondence

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

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

Prerequisites

• Triangle — First establish three distinct noncollinear points.
• Euclidean distance in an equal-unit orthonormal frame — Compare corresponding side lengths through their squares.
• Determinants of real 2×2 matrices — 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.