W.16 The Euler Line & Nine-Point Circle Over Arbitrary Fields — Code

In any non-null triangle A_1 A_2 A_3 over a field F (char F != 2), the nine-point center N = (C + O)/2 satisfies 2*N - C = O (and 3*G = 2*C + O when char F != 3), placing C, G, N, O on the Euler line with quadrance proportions Q(C, G) : Q(C, N) : Q(C, O) = 4 : 9 : 36; the three side midpoints M_i, three altitude feet F_i, and three Euler midpoints E_i = (A_i + O)/2 all have exact quadrance K / 4 from N.

Exact geometor.model SVG output of model.py for W.16 (The Euler Line & Nine-Point Circle Over Arbitrary Fields).

Symbolic Construction & Verification Script (model.py)

"""GEOMETOR Wildberger symbolic model and proof script for W.16: The Euler Line & Nine-Point Circle Over Arbitrary Fields.

        Enunciation: In any non-null triangle A_1 A_2 A_3 over a field F (char F != 2), the nine-point center N = (C + O)/2 satisfies 2*N - C = O (and 3*G = 2*C + O when char F != 3), placing C, G, N, O on the Euler line with quadrance proportions Q(C, G) : Q(C, N) : Q(C, O) = 4 : 9 : 36; the three side midpoints M_i, three altitude feet F_i, and three Euler midpoints E_i = (A_i + O)/2 all have exact quadrance K / 4 from N.
        """

        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_16() -> TopicBuildResult:
            """Construct the exact symbolic model and verifications for W.16.

            Returns:
                Populated `TopicBuildResult` for W.16 (The Euler Line & Nine-Point Circle Over Arbitrary Fields).
            """
            result = build_principle_model("W.16")
            assert_zero_floats(result.model)
            return result


        def main() -> None:
            """Execute W.16 symbolic construction, render `w-16.svg`, and print invariants."""
            here = Path(__file__).resolve().parent
            bundle = construct_w_16()
            svg_out = render_svg(
                bundle.model,
                here / "w-16.svg",
                extra_curves=bundle.extra_curves,
                annotations=bundle.annotations,
            )
            print("W.16: The Euler Line & Nine-Point Circle Over Arbitrary Fields")
            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()