G.588 equilateral pentagon inscribed — Code

XIII.10

If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on the side of the hexagon and on that of the decagon inscribed in the same circle.

"""Programmatic construction for Book XIII, Proposition 10 (XIII.10).

        Regular pentagon ABCDE inscribed in circle ABCDE with centre F, diameter AG,
        perpendiculars FHK to AB and FLM to AK, and chord KN.
        """

        from __future__ import annotations

        import math
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0462.g1 using geometor.model."""
            model = Model("XIII.10")

            cx, cy, r = 151.0, -154.0, 120.0
            F = model.set_point(cx, cy, classes=["given"], ID="F")

            # Regular pentagon vertices A (top), B (left), C (bottom-left), D (bottom-right), E (right)
            angles_deg = [90.0, 162.0, -126.0, -54.0, 18.0]
            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"]

            # Circle ABCDE
            model.construct_circle(F, A)

            # Diameter point G at bottom (-90 deg)
            G = model.set_point(151.0, -274.0, classes=["constructed"], ID="G")

            # K bisects arc AB (126 deg), H is midpoint of AB
            K = model.set_point(80.5, -56.9, classes=["constructed"], ID="K")
            H = model.set_point(93.9, -75.5, classes=["constructed"], ID="H")

            # M bisects arc AK (108 deg), L is midpoint of AK, N is intersection of FM and AB
            M = model.set_point(113.9, -39.9, classes=["constructed"], ID="M")
            L = model.set_point(115.7, -45.5, classes=["constructed"], ID="L")
            N = model.set_point(119.5, -56.9, classes=["constructed"], ID="N")

            # Polygons and triangles
            model.set_polygon([A, B, C, D, E])
            model.set_polygon([A, B, F])
            model.set_polygon([B, F, N])
            model.set_polygon([A, K, B])
            model.set_polygon([A, K, N])

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

            # Sections
            model.set_section([A, F, G])
            model.set_section([A, N, B])
            model.set_section([F, H, K])
            model.set_section([A, L, K])
            model.set_section([F, N, M])

            # Angles (wedges)
            model.set_wedge(F, A, A, K)
            model.set_wedge(F, K, K, B)
            model.set_wedge(F, G, G, B)
            model.set_wedge(F, B, B, M)
            model.set_wedge(A, F, F, B)
            model.set_wedge(B, A, A, F)
            model.set_wedge(F, A, A, B)
            model.set_wedge(N, B, B, F)
            model.set_wedge(K, L, L, N)
            model.set_wedge(A, L, L, N)
            model.set_wedge(B, K, K, N)
            model.set_wedge(K, A, A, B)
            model.set_wedge(N, K, K, A)

            return model


        if __name__ == "__main__":
            construct_g1()