G.589 circle diameter rational — Code

XIII.11

If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational segment called minor.

"""Programmatic construction for Book XIII, Proposition 11 (XIII.11).

        Regular pentagon ABCDE inscribed in circle ABCDE with rational diameter AG,
        centre F, diameter BH meeting AC at M, AG meeting CD at L, FK = 1/4 AF,
        and auxiliary line N.
        """

        from __future__ import annotations

        import math
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0466.g1 using geometor.model."""
            model = Model("XIII.11")

            cx, cy, r = 204.0, -152.0, 122.0
            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 endpoints G (-90 deg) and H (-18 deg)
            G = model.set_point(204.0, -274.0, classes=["constructed"], ID="G")
            H = model.set_point(320.0, -189.7, classes=["constructed"], ID="H")

            # L (intersection of AG and CD) and M (intersection of BH and AC)
            L = model.set_point(204.0, -250.7, classes=["constructed"], ID="L")
            M = model.set_point(168.1, -140.4, classes=["constructed"], ID="M")

            # K on FH with FK = 1/4 AF
            K = model.set_point(233.0, -161.4, classes=["constructed"], ID="K")

            # Auxiliary line N on the left
            N1 = model.set_point(24.0, -76.0, classes=["auxiliary"], ID="N1")
            N2 = model.set_point(24.0, -216.0, classes=["auxiliary"], ID="N2")
            model.set_segment(N1, N2)

            # Polygons and triangles
            model.set_polygon([A, B, C, D, E])
            model.set_polygon([A, C, L])
            model.set_polygon([A, M, F])
            model.set_polygon([A, B, H])
            model.set_polygon([A, B, M])

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

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

            # Angles (wedges)
            model.set_wedge(L, A, A, C)
            model.set_wedge(M, A, A, F)
            model.set_wedge(A, L, L, C)
            model.set_wedge(C, A, A, L)
            model.set_wedge(F, M, M, A)

            return model


        if __name__ == "__main__":
            construct_g1()