G.93 cut a segment in extreme and mean ratio — Code

II.11

Cut a given segment so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

"""Programmatic construction for Book II, Proposition 11 — The Golden Section (II.11).

        To cut a given straight line so that the rectangle contained by the whole and
        one of the segments is equal to the square on the remaining segment.
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct Euclid's Golden Section figure (Book II, Prop. 11) using geometor.model."""
            model = Model("II.11")

            # 1. Let AB be the given straight line (length 2 for clean midpoint E).
            A = model.set_point(0, 0, classes=["given"], ID="A")
            B = model.set_point(2, 0, classes=["given"], ID="B")
            model.set_segment(A, B)

            # 2. Let the square ABDC be described on AB [I. 46].
            C = model.set_point(0, -2, ID="C")
            D = model.set_point(2, -2, ID="D")
            model.set_polygon([A, B, D, C])

            # 3. Let AC be bisected at E, and let BE be joined.
            E = model.set_point(0, -1, ID="E")
            model.set_segment(B, E)

            # 4. Let CA be drawn through to F, and let EF be made equal to BE (sqrt(5)).
            model.construct_line(C, A)
            phi_inv_2 = sp.sqrt(5) - 1  # AF = AH = sqrt(5) - 1 (for AB = 2)
            F = model.set_point(0, phi_inv_2, ID="F")

            # 5. Let the square FH (FGHA) be described on AF, and let GH be drawn through to K.
            G = model.set_point(phi_inv_2, phi_inv_2, ID="G")
            H = model.set_point(phi_inv_2, 0, classes=["golden"], ID="H")
            K = model.set_point(phi_inv_2, -2, ID="K")
            model.construct_line(G, K)

            # 6. Record the Golden Sections on AB and CF
            model.set_section([A, H, B], classes=["golden"])
            model.set_section([C, A, F], classes=["golden"])

            # 7. Explicitly add all referenced segments
            model.set_segment(A, H)
            model.set_segment(H, B)
            model.set_segment(A, E)
            model.set_segment(E, C)
            model.set_segment(A, C)
            model.set_segment(A, F)
            model.set_segment(E, F)
            model.set_segment(C, F)
            model.set_segment(F, G)
            model.set_segment(G, H)
            model.set_segment(H, K)
            model.set_segment(G, K)
            model.set_segment(B, D)
            model.set_segment(C, K)
            model.set_segment(K, D)
            model.set_segment(C, D)

            # 8. Explicitly add all referenced squares, rectangles, triangles, and right angle
            model.set_polygon([F, G, H, A])  # Square FH on AF
            model.set_polygon([F, G, K, C])  # Rectangle FK (CF, FA)
            model.set_polygon([A, H, K, C])  # Common rectangle AK
            model.set_polygon([H, B, D, K])  # Remainder rectangle HD (AB, BH)
            model.set_polygon([A, B, E])     # Right triangle ABE
            model.set_wedge(A, B, B, E)      # Right angle at A

            return model


        if __name__ == "__main__":
            construct_g1()