G.586 equilateral equiangular pentagon — Code

XIII.8

If in an equilateral and equiangular pentagon segments subtend two angles taken in order, they cut one another in extreme and mean ratio, and their greater segments are equal to the side of the pentagon.

"""Programmatic construction for Book XIII, Proposition 8 (XIII.8).

        An equilateral and equiangular pentagon ABCDE inscribed in circle ABCDE,
        with diagonals AC and BE intersecting at H in extreme and mean ratio.
        """

        from __future__ import annotations

        import math
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0457.g1 using geometor.model."""
            model = Model("XIII.8")

            cx, cy, r = 108.0, -108.0, 86.0
            O = model.set_point(cx, cy, classes=["auxiliary"], ID="O")

            # Regular pentagon vertices A (top), B (right), C (bottom-right), D (bottom-left), E (left)
            angles_deg = [90.0, 18.0, -54.0, -126.0, 162.0]
            labels = ["A", "B", "C", "D", "E"]
            pts = {}
            for lbl, deg in zip(labels, angles_deg):
                rad = math.radians(deg)
                pts[lbl] = model.set_point(
                    round(cx + r * math.cos(rad), 1),
                    round(cy + r * math.sin(rad), 1),
                    classes=["given"],
                    label=lbl,
                )

            A, B, C, D, E = pts["A"], pts["B"], pts["C"], pts["D"], pts["E"]

            # Circumscribed circle ABCDE
            model.construct_circle(O, A)

            # Intersection H of diagonals AC and BE
            H = model.set_point(127.3, -81.4, classes=["constructed"], ID="H")

            # Pentagon and triangles
            model.set_polygon([A, B, C, D, E])
            model.set_polygon([A, B, E])
            model.set_polygon([A, B, C])
            model.set_polygon([A, B, H])

            # Segments and collinear sub-segments
            for p1, p2 in [
                (A, B), (B, C), (C, D), (D, E), (E, A),
                (A, C), (A, H), (H, C),
                (B, E), (E, H), (H, B),
            ]:
                model.set_segment(p1, p2)

            # Sections
            model.set_section([A, H, C])
            model.set_section([E, H, B])

            # Angles (wedges)
            model.set_wedge(A, B, B, C)
            model.set_wedge(B, A, A, E)
            model.set_wedge(H, A, A, E)
            model.set_wedge(A, E, E, C)
            model.set_wedge(E, A, A, B)
            model.set_wedge(A, B, B, E)
            model.set_wedge(H, A, A, B)

            return model


        if __name__ == "__main__":
            construct_g1()