W.09 Law IV: The Cross Law & Quadrea-Spread Identity — Code

In any triangle A_1 A_2 A_3 with non-null sides Q_1, Q_2 meeting with spread s_3 and cross c_3 = 1 - s_3 at vertex A_3, the quadrances satisfy (Q_1 + Q_2 - Q_3)^2 = 4 * Q_1 * Q_2 * (1 - s_3) = 4 * Q_1 * Q_2 * c_3, equivalently A(Q_1, Q_2, Q_3) = 4 * Q_1 * Q_2 * s_3.

Exact geometor.model SVG output of model.py for W.09 (Law IV: The Cross Law & Quadrea-Spread Identity).

Symbolic Construction & Verification Script (model.py)

"""GEOMETOR Wildberger symbolic model and proof script for W.09: Law IV: The Cross Law & Quadrea-Spread Identity.

        Enunciation: In any triangle A_1 A_2 A_3 with non-null sides Q_1, Q_2 meeting with spread s_3 and cross c_3 = 1 - s_3 at vertex A_3, the quadrances satisfy (Q_1 + Q_2 - Q_3)^2 = 4 * Q_1 * Q_2 * (1 - s_3) = 4 * Q_1 * Q_2 * c_3, equivalently A(Q_1, Q_2, Q_3) = 4 * Q_1 * Q_2 * s_3.
        """

        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_09() -> TopicBuildResult:
            """Construct the exact symbolic model and verifications for W.09.

            Returns:
                Populated `TopicBuildResult` for W.09 (Law IV: The Cross Law & Quadrea-Spread Identity).
            """
            result = build_principle_model("W.09")
            assert_zero_floats(result.model)
            return result


        def main() -> None:
            """Execute W.09 symbolic construction, render `w-09.svg`, and print invariants."""
            here = Path(__file__).resolve().parent
            bundle = construct_w_09()
            svg_out = render_svg(
                bundle.model,
                here / "w-09.svg",
                extra_curves=bundle.extra_curves,
                annotations=bundle.annotations,
            )
            print("W.09: Law IV: The Cross Law & Quadrea-Spread Identity")
            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()