Geometry Solver — Triangle mechanism

Model scope

The listed law-of-cosines and Heron formulas are standard and correct under their stated correspondence and validity constraints. The specification is not complete enough to verify the full claimed instrument because formulas for perimeter, altitude, inradius, circumradius, and classification are absent, and angle-unit handling and SSA branch behavior are not fully defined. The honest label correctly identifies the SSA ambiguous case, but the UI and output contract should explicitly show zero, one, or two solutions.

• The law of cosines formula is correct when side c is opposite angle C, but the specification must define the corresponding permutations for other sides.
• The law of sines relation is correct but incomplete as a solver specification; it does not state how to compute all remaining sides and angles or resolve inverse-sine branches.
• Heron's formula is correct for a valid triangle, subject to the triangle inequalities and nonnegative radicand.
• Formulas are missing for the claimed perimeter, altitude, inradius, circumradius, and classification outputs.
• The input description '3 known sides/angles' is ambiguous: solvability and uniqueness depend on whether the case is SSS, SAS, ASA, AAS, or SSA.
• SSA must explicitly enumerate zero, one, or two valid solutions and reject invalid inverse-sine branches.

Units and conventions

• All side lengths use the same linear unit; perimeter, altitudes, inradius, and circumradius use that unit, while area uses its square.
• Angles in the test vectors are degrees; implementations using standard JavaScript trigonometric functions must convert degrees to radians before evaluating sin, cos, or inverse trigonometric functions.
• Angles must be interpreted consistently as interior angles in the range 0° < angle < 180° for nondegenerate triangles.
• The law of cosines requires each side-angle correspondence to be correct, such as side c opposite angle C.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "case": "SSS",
    "a": 3,
    "b": 4,
    "c": 5,
    "length_unit": "cm"
  },
  "expected_outputs": {
    "angles": {
      "A": "36.86989765°",
      "B": "53.13010235°",
      "C": "90°"
    },
    "area": "6 cm²",
    "perimeter": "12 cm",
    "altitudes": {
      "h_a": "4 cm",
      "h_b": "3 cm",
      "h_c": "2.4 cm"
    },
    "inradius": "1 cm",
    "circumradius": "2 cm",
    "classification": "right"
  },
  "working": "s=(3+4+5)/2=6. Heron area=√(6(6−3)(6−4)(6−5))=√(6×3×2×1)=√36=6 cm². Perimeter=3+4+5=12 cm. h_a=2K/a=2×6/3=4 cm; h_b=2×6/4=3 cm; h_c=2×6/5=2.4 cm. r=K/s=6/6=1 cm. R=abc/(4K)=(3×4×5)/(4×6)=60/24=2 cm. cos C=(a²+b²−c²)/(2ab)=(9+16−25)/(24)=0, so C=90°; A=arcsin(3/5)=36.86989765° and B=90°−A=53.13010235°."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The instrument reaches undefined behavior for zero-angle SSA input, must reject exact triangle-inequality degeneracy, and has a major unhandled unit-convention ambiguity for numeric angles. The convention vector also exposes inconsistent intermediate arithmetic: the correct degree SAS results are c=1.001826 and area=0.104714.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json