G.601 set out sides — Code

XIII.18

To set out the sides of the five figures and to compare them with one another.

"""Programmatic construction for Book XIII, Proposition 18 (XIII.18).

        Comparison of the sides of the five regular solids inscribed in a given sphere
        with diameter AB and semicircle AEB.
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0507.g1 using geometor.model."""
            model = Model("XIII.18")

            # Given diameter AB with center C
            A = model.set_point(-1, 0, classes=["given"], ID="A")
            B = model.set_point(1, 0, classes=["given"], ID="B")
            C = model.set_point(0, 0, classes=["given"], ID="C")

            # D divides AB such that AD = 2 DB -> D is at x = 1/3
            D = model.set_point(sp.Rational(1, 3), 0, classes=["constructed"], label="D")

            # Semicircle AEB on diameter AB with center C
            model.construct_circle(C, A)

            # CE perpendicular to AB at C -> E = (0, 1)
            E = model.set_point(0, 1, classes=["constructed"], ID="E")

            # DF perpendicular to AB at D -> F = (1/3, sqrt(8)/3)
            F = model.set_point(sp.Rational(1, 3), sp.sqrt(8) / 3, classes=["constructed"], label="F")

            # AG perpendicular to AB at A, AG = AB = 2 -> G = (-1, 2)
            G = model.set_point(-1, 2, classes=["constructed"], ID="G")

            # GC meets semicircle at H = (-1/sqrt(5), 2/sqrt(5))
            H = model.set_point(-1 / sp.sqrt(5), 2 / sp.sqrt(5), classes=["constructed"], label="H")

            # HK perpendicular to AB at K = (-1/sqrt(5), 0)
            K = model.set_point(-1 / sp.sqrt(5), 0, classes=["constructed"], label="K")

            # CL = CK -> L = (1/sqrt(5), 0)
            L = model.set_point(1 / sp.sqrt(5), 0, classes=["constructed"], label="L")

            # LM perpendicular to AB at L -> M = (1/sqrt(5), 2/sqrt(5))
            M = model.set_point(1 / sp.sqrt(5), 2 / sp.sqrt(5), classes=["constructed"], label="M")

            # N divides FB in extreme and mean ratio with NB as the greater segment
            phi_inv = (sp.sqrt(5) - 1) / 2
            N = model.set_point(
                B.x + phi_inv * (F.x - B.x),
                B.y + phi_inv * (F.y - B.y),
                classes=["constructed"],
                label="N",
            )

            # Base lines
            model.construct_line(A, B)
            model.construct_line(A, G)
            model.construct_line(G, C)

            # Polygons (triangles)
            model.set_polygon([A, F, B])
            model.set_polygon([A, F, D])
            model.set_polygon([F, D, B])

            # Segments along diameter AB
            for p1, p2 in [
                (A, B), (A, C), (C, B), (A, D), (D, B),
                (C, D), (C, K), (C, L), (K, L), (A, K), (L, B), (A, L),
            ]:
                model.set_segment(p1, p2)

            # Vertical and oblique segments
            for p1, p2 in [
                (C, E), (D, F), (A, G), (H, K), (L, M),
                (A, E), (E, B), (A, F), (F, B), (F, N), (N, B),
                (A, M), (M, B), (G, C), (G, H), (H, C),
            ]:
                model.set_segment(p1, p2)

            return model


        if __name__ == "__main__":
            construct_g1()