W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea — Code

Three planar points A_1, A_2, A_3 with pairwise quadrances Q_1, Q_2, Q_3 are collinear if and only if Archimedes’ symmetric quadratic form A(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2) vanishes; for any triangle, A(Q_1, Q_2, Q_3) equals 4 times the squared coordinate determinant (16 * Delta^2), defining the triangle’s quadrea.

Exact geometor.model SVG output of model.py for W.07 (Law II: Triple Quad Formula, Archimedes’ Function & Quadrea).

Symbolic Construction & Verification Script (model.py)

"""GEOMETOR Wildberger symbolic model and proof script for W.07: Law II: Triple Quad Formula, Archimedes' Function & Quadrea.

        Enunciation: Three planar points A_1, A_2, A_3 with pairwise quadrances Q_1, Q_2, Q_3 are collinear if and only if Archimedes' symmetric quadratic form A(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2) vanishes; for any triangle, A(Q_1, Q_2, Q_3) equals 4 times the squared coordinate determinant (16 * Delta^2), defining the triangle's quadrea.
        """

        from __future__ import annotations

        import sys
        from pathlib import Path

        _ROOT_SRC = Path(__file__).resolve().parents[3] / "src"
        if str(_ROOT_SRC) not in sys.path:
            sys.path.insert(0, str(_ROOT_SRC))

        from geometor.wildberger.models import (  # noqa: E402
            TopicBuildResult,
            assert_zero_floats,
            build_principle_model,
        )
        from geometor.wildberger.render import render_svg  # noqa: E402


        def construct_w_07() -> TopicBuildResult:
            """Construct the exact symbolic model and verifications for W.07.

            Returns:
                Populated `TopicBuildResult` for W.07 (Law II: Triple Quad Formula, Archimedes' Function & Quadrea).
            """
            result = build_principle_model("W.07")
            assert_zero_floats(result.model)
            return result


        def main() -> None:
            """Execute W.07 symbolic construction, render `w-07.svg`, and print invariants."""
            here = Path(__file__).resolve().parent
            bundle = construct_w_07()
            svg_out = render_svg(
                bundle.model,
                here / "w-07.svg",
                extra_curves=bundle.extra_curves,
                annotations=bundle.annotations,
            )
            print("W.07: Law II: Triple Quad Formula, Archimedes' Function & Quadrea")
            print(f"  Elements in model: {len(bundle.model)}")
            print(f"  Rendered SVG:      {svg_out.name}")
            for label, val, status in bundle.verifications:
                print(f"  [OK] {label}: {val} ({status})")


        if __name__ == "__main__":
            main()