Symbolic Demonstrations

Executable pure-symbolic Python demonstrations (Model(…, auto_intersect=False)) verifying every static lever equilibrium, signed barycentric coordinate, isotomic conjugate, projective harmonic range, Rational Trigonometry spread law, multilateral odd-polygon product, 3D seven-line tetrahedral concurrency, and Appendix Geometrica quadrature/cubature in geometor.ceva with zero floating-point coordinates.

1. Static Barycentrics, Negative Weights (F - A), Isotomic Conjugates & Spread Ceva

Proves Ceva’s Theorem and sub-triangle area proportions universally over indeterminate coordinates, evaluates canonical centers $X(1), X(2), X(4), X(6), X(7), X(8)$ and the exterior $A$-excenter $J_A$ over $\mathbb{Q}^2$ on the Heronian $(13, 14, 15)$ triangle, constructs @proposition models for interior/exterior concurrency, isotomic conjugation ($X(7) \leftrightarrow X(8)$), and the Menelaus harmonic conjugate $(B, C; D, D') = -1$, and verifies Spread Ceva’s Theorem without square roots.

#!/usr/bin/env python3
        """Exact symbolic demonstration of Ceva's static barycentrics, isotomic conjugates, and Spread Ceva."""
        
        from __future__ import annotations
        
        import sympy as sp
        from geometor.model import cross_ratio, is_harmonic
        from rich.console import Console
        from rich.table import Table
        
        from geometor.ceva import (
            CANONICAL_CENTERS,
            construct_ceva_concurrency,
            construct_cevian_menelaus_harmonic,
            construct_isotomic_conjugate,
            evaluate_canonical_centers,
            prove_ceva_barycentric_identity,
            verify_spread_ceva,
        )
        
        
        def run_demo() -> None:
            """Execute exact symbolic proofs for Liber Primus and canonical triangle centers."""
            console = Console()
            console.print(
                "[bold amber1]1. Universal Symbolic Proof of Ceva's Barycentric & Area Identities[/bold amber1]"
            )
            residuals = prove_ceva_barycentric_identity()
            for key, expr in residuals.items():
                assert expr == 0, f"Non-zero symbolic residual {key}: {expr}"
                console.print(f"  • {key}: [green]{expr}[/green]")
        
            console.print(
                "\n[bold amber1]2. Canonical Rational Centers on Heronian Δ(13, 14, 15)[/bold amber1]"
            )
            tbl = Table(show_lines=False)
            tbl.add_column("Center", style="bold cyan")
            tbl.add_column("Weights (u:v:w)", style="magenta")
            tbl.add_column("Exact P in Q^2", style="green")
            tbl.add_column("Signed Ratios (AF/FB, BD/DC, CE/EA)", style="yellow")
            tbl.add_column("Product")
        
            for code, res in evaluate_canonical_centers().items():
                _desc, (u_val, v_val, w_val) = CANONICAL_CENTERS[code]
                tbl.add_row(
                    code,
                    f"{u_val}:{v_val}:{w_val}",
                    f"({res.center.x}, {res.center.y})",
                    f"({res.ratio_af_fb}, {res.ratio_bd_dc}, {res.ratio_ce_ea})",
                    str(res.ceva_product),
                )
            console.print(tbl)
        
            console.print(
                "\n[bold amber1]3. Constructive @proposition Models (Interior, Exterior F-A, Isotomic, Harmonic)[/bold amber1]"
            )
            m_int = construct_ceva_concurrency(24, 28, 21)
            m_ext = construct_ceva_concurrency(-14, 15, 13)
            m_iso = construct_isotomic_conjugate(24, 28, 21)
            m_har = construct_cevian_menelaus_harmonic(24, 28, 21)
        
            b_pt = m_har.get_element_by_ID("B")
            c_pt = m_har.get_element_by_ID("C")
            d_pt = m_har.get_element_by_ID("D")
            dp_pt = m_har.get_element_by_ID("D_prime")
            cr_val = cross_ratio(b_pt, c_pt, d_pt, dp_pt)
            assert is_harmonic(b_pt, c_pt, d_pt, dp_pt)
        
            console.print(f"  • Interior Gergonne X(7) Model elements: [green]{len(m_int)}[/green]")
            console.print(f"  • Exterior Excenter J_A (-14:15:13) elements: [green]{len(m_ext)}[/green]")
            console.print(f"  • Isotomic X(7) <-> X(8) Model elements: [green]{len(m_iso)}[/green]")
            console.print(
                f"  • Cevian–Menelaus Cross-Ratio (B, C; D, D'): [green]{sp.simplify(cr_val)}[/green]"
            )
        
            console.print(
                "\n[bold amber1]4. Rational Trigonometry Spread Ceva Theorem (Zero Radicals)[/bold amber1]"
            )
            spread = verify_spread_ceva()
            console.print(
                f"  • s(BAD)={spread.spread_bad}, s(DAC)={spread.spread_dac} | "
                f"s(CBE)={spread.spread_cbe}, s(EBA)={spread.spread_eba} | "
                f"s(ACF)={spread.spread_acf}, s(FCB)={spread.spread_fcb}"
            )
            console.print(
                f"  • Spread Product = [green]{spread.spread_product}[/green], "
                f"Quadrance Product = [green]{spread.quadrance_product}[/green]"
            )
        
        
        if __name__ == "__main__":
            run_demo()
        

Run locally:

/home/phi/PROJECTS/geometor/.venv/bin/python demos/static_barycentrics.py
        

2. Multilateral Odd Polygons, Circumscribed Conics, 3D Tetrahedra & Appendix Geometrica

Constructs concurrent cevians on a convex rational pentagon, proves Liber Secundus, Lemma I symbolically for polygons circumscribed around circles and Borelli–Serenus projected ellipses, verifies Ceva’s 3D skew-quadrilateral corollary (Elementum V Coroll.) and 7-line tetrahedral equilibrium (Liber Primus, Prop. XXIII), and confirms Hippocratean lune quadrature, cylindrical ungula $\frac{2}{3}$ prism cubature, and hemisphere ($5:3$) / paraboloid ($2:1$) centroids.

#!/usr/bin/env python3
        """Exact symbolic demonstration of multilateral polygons, 3D tetrahedra, and Appendix Geometrica."""
        
        from __future__ import annotations
        
        from rich.console import Console
        
        from geometor.ceva import (
            construct_pentagon_ceva,
            prove_circumscribed_polygon_lemma,
            solve_tetrahedron_static_equilibrium,
            verify_appendix_invariants,
            verify_odd_polygon_ceva,
            verify_skew_quadrilateral_ceva,
        )
        
        
        def run_demo() -> None:
            """Execute exact symbolic verifications for Liber Secundus, 3D statics, and Appendix."""
            console = Console()
        
            console.print(
                "[bold amber1]1. Liber Secundus: Odd Multilateral Polygon Ceva & Circumscribed Conic Lemma I[/bold amber1]"
            )
            pent = verify_odd_polygon_ceva()
            pent_model = construct_pentagon_ceva()
            conic_lemma = prove_circumscribed_polygon_lemma(5)
            console.print(f"  • Rational Pentagon side ratios: {pent.side_ratios}")
            console.print(
                f"  • Cyclic opposite-side product = [green]{pent.cyclic_product}[/green] "
                f"(Model elements: {len(pent_model)})"
            )
            console.print(f"  • Borelli–Serenus Circumscribed Ellipse residuals: {conic_lemma}")
        
            console.print(
                "\n[bold amber1]2. Liber Primus Elem. V Coroll. & Prop. XXIII: 3D Skew Quad & 7-Line Tetrahedron[/bold amber1]"
            )
            skew = verify_skew_quadrilateral_ceva()
            tet = solve_tetrahedron_static_equilibrium()
            console.print(
                f"  • 3D Skew Quadrilateral: 6·Vol = {skew.tetrahedron_det6}, "
                f"coplanarity det = [green]{skew.fulcrum_coplanarity_det}[/green], "
                f"cyclic product = [green]{skew.cyclic_ratio_product}[/green]"
            )
            console.print(
                f"  • 3D Tetrahedron Center O = [green]{tuple(tet.center_of_gravity)}[/green]"
            )
            console.print(f"  • 4 Vertex-to-Face-CG ratios VO/OG: {tet.vertex_cevian_ratios}")
            console.print(f"  • 3 Skew-Edge Bimedian ratios: {tet.bimedian_ratios}")
        
            console.print(
                "\n[bold amber1]3. Appendix Geometrica: Lune Quadrature, Ungula Cubature & Centroids[/bold amber1]"
            )
            app = verify_appendix_invariants()
            for k, val in app.items():
                console.print(f"  • {k}: [green]{val}[/green]")
        
        
        if __name__ == "__main__":
            run_demo()
        

Run locally:

/home/phi/PROJECTS/geometor/.venv/bin/python demos/polygon_tetrahedron.py