F.36 Orthic Triangle — Code

The orthic triangle H_A H_B H_C formed by the altitude feet of an acute triangle ABC is the pedal triangle of the orthocenter H = X(4), is homothetic to the tangential triangle, has H as its incenter, and solves Fagnano’s 1775 problem by minimizing the perimeter among all triangles inscribed in ABC.

Output of model.py for F.36 (Orthic Triangle).

Symbolic Construction Script (model.py)

"""GEOMETOR Fundamentals construction model for F.36: Orthic Triangle.

        Enunciation:
            The orthic triangle H_A H_B H_C formed by the altitude feet of an acute triangle ABC is the pedal triangle of the orthocenter H = X(4), is homothetic to the tangential triangle, has H as its incenter, and solves Fagnano's 1775 problem by minimizing the perimeter among all triangles inscribed in ABC.
        """
        from __future__ import annotations

        from pathlib import Path

        import sympy as sp
        from geometor.fundamentals.construct import build_topic_model
        from geometor.fundamentals.render import render_svg
        from geometor.model import Model


        def construct_f_36() -> tuple[Model, dict[str, sp.Expr]]:
            """Construct the exact symbolic/geometric model for F.36 (Orthic Triangle).

            Returns:
                Tuple of the populated `geometor.model.Model` and a dictionary of
                symbolic SymPy invariants for the reference configuration.
            """
            model, _ = build_topic_model(36, 'Orthic Triangle')

            # Canonical Seelig reference triangle (a, b, c) = (5, 6, 7) symbolic invariants
            a, b, c = sp.Integer(5), sp.Integer(6), sp.Integer(7)
            s = (a + b + c) / 2
            area = sp.sqrt(s * (s - a) * (s - b) * (s - c))
            inradius = area / s
            circumradius = (a * b * c) / (4 * area)
            s_a = (b**2 + c**2 - a**2) / 2
            s_b = (c**2 + a**2 - b**2) / 2
            s_c = (a**2 + b**2 - c**2) / 2

            invariants: dict[str, sp.Expr] = {
                "s": s,
                "K": area,
                "r": sp.simplify(inradius),
                "R": sp.simplify(circumradius),
                "S_A": s_a,
                "S_B": s_b,
                "S_C": s_c,
            }
            return model, invariants


        if __name__ == "__main__":
            here = Path(__file__).resolve().parent
            model, extra_curves = build_topic_model(36, 'Orthic Triangle')
            svg_out = render_svg(model, here / "f-36.svg", extra_curves=extra_curves)
            _, sym_invariants = construct_f_36()
            print(f"F.36: Orthic Triangle")
            print(f"  Elements in model: {len(model)}")
            print(f"  Rendered SVG:      {svg_out.name}")
            for k, v in sym_invariants.items():
                print(f"  {k} = {v}")