Labeled triangle similarity with a specified positive scale

Mathematical statement

For two nondegenerate Euclidean plane triangles with explicit correspondence Ai↔Bi, the SSS similarity slice at a specified positive scale k requires |BiBj|=k|AiAj| for every corresponding side. Equivalently, all three squared side lengths scale by k². If a positive area of A is given, the area of B is k² times that value. Reflection does not change the positive length scale.

Representative formula (exact TeX)

S(A,B)

Assumptions

• Both triangles are nondegenerate in one equal-unit orthonormal frame. Correspondence and scale direction are explicit.
• Exercises use k=p/q with q>0 and, except for scale-eligibility questions, p>0. Eligibility questions allow p≤0 and reject those candidates.
• Failure at the specified k rejects only that scale witness, not every possible similarity scale. Area questions provide a positive rational area and do not claim that the rectangle-area object proves general triangle-area theory.

Symbol scope

• S — labeled similarity relation at a specified positive scale
• A — source triangle for the scale
• B — target triangle for the scale

Intuition, definition, and misconceptions

Lengths scale by k, and squared lengths scale by k². Nondegeneracy makes side lengths positive, permitting the exact integer cross-product check B²q²=A²p².

Area exercises are conditional calculations with given similar triangles, a positive scale, and a known positive area. Do not guess area from coordinate sketches or extend this to general measure theory.

Uniform coordinate scaling by −2 still gives positive length scale two. A negative coordinate coefficient and a negative length scale are different notions. Scale zero is degenerate and outside this nondegenerate similarity slice.

Common errors include scaling both length and area by k, confusing squared scale with length scale, or using failure at a specified k to reject every possible similarity.

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

• Congruence of nondegenerate triangles with explicit correspondence — Retain nondegeneracy and corresponding-distance decisions.
• Ratio and proportion — Use positive rational scale and the specified ratio direction.
• Natural-number powers and principal square roots — Use squared lengths and the square of the scale.

Verification boundary

Positive rational-scale instances can be recomputed exactly; general similarity maps, SSS, and triangle-area ratios have no attached formal Evidence.

not_formalized

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

Next obligation

Bind corresponding distances at a positive scale and the k² area ratio to the pinned version, without borrowing absent Evidence from the rectangle or cell model.

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.