W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points — Code

On a circle Q(X, O_c) = K over F, a diameter subtends spread 1 at every other circle point (Thales), any chord P_1 P_2 subtends constant spread s_3 at the circumference with Q(P_1, P_2) = 4*K*s_3 and central spread s(O_c P_1, O_c P_2) = S_2(s_3) = 4*s_3*(1 - s_3), and any external or internal point A_0 has invariant secant quadrance product Q(A_0, P_1)*Q(A_0, P_2) = (Q(A_0, O_c) - K)^2.
c (Q = 25) / P_1 P_2 / (Q = 50) / B P_1 / (Q = 90) / B P_2 / (Q = 80) l_sec1 l_sec2 O_c = [0, 0] P_1 = [-3, 4] P_2 = [4, 3] B = [0, -5] Q_1 = [5, 0] Q_2 = [4, -3] A_0 = [15/2, -5/4] s(BP_1,BP_2) = 1/2, central = S_2(s) = 1 Secant Q-prod = 275625/256 OcOc P1P1 P2P2 BB Q1Q1 Q2Q2 A0A0 + − ⟲ Lbl Gde

Exact geometor.model SVG output of model.py for W.17 (Circle Geometry: Subtended Spreads, Secant Products & Rational Points).

Symbolic Construction & Verification Script (model.py)

"""GEOMETOR Wildberger symbolic model and proof script for W.17: Circle Geometry: Subtended Spreads, Secant Products & Rational Points.

        Enunciation: On a circle Q(X, O_c) = K over F, a diameter subtends spread 1 at every other circle point (Thales), any chord P_1 P_2 subtends constant spread s_3 at the circumference with Q(P_1, P_2) = 4*K*s_3 and central spread s(O_c P_1, O_c P_2) = S_2(s_3) = 4*s_3*(1 - s_3), and any external or internal point A_0 has invariant secant quadrance product Q(A_0, P_1)*Q(A_0, P_2) = (Q(A_0, O_c) - K)^2.
        """

        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_17() -> TopicBuildResult:
            """Construct the exact symbolic model and verifications for W.17.

            Returns:
                Populated `TopicBuildResult` for W.17 (Circle Geometry: Subtended Spreads, Secant Products & Rational Points).
            """
            result = build_principle_model("W.17")
            assert_zero_floats(result.model)
            return result


        def main() -> None:
            """Execute W.17 symbolic construction, render `w-17.svg`, and print invariants."""
            here = Path(__file__).resolve().parent
            bundle = construct_w_17()
            svg_out = render_svg(
                bundle.model,
                here / "w-17.svg",
                extra_curves=bundle.extra_curves,
                annotations=bundle.annotations,
            )
            print("W.17: Circle Geometry: Subtended Spreads, Secant Products & Rational Points")
            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()