G.116 diameter is the greatest chord — Code

III.15

Of chords in a circle, the diameter is the greatest, and a chord nearer the center is always greater than one more remote.

"""Programmatic construction for Book III, Proposition 15 (III.15).

        Of straight lines in a circle the diameter is greatest, and of the rest the
        nearer to the centre is always greater than the more remote.
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure for Book III, Proposition 15 using geometor.model."""
            model = Model("III.15")

            # 1. Circle ABCD about centre E with diameter AD
            E = model.set_point(0, 0, classes=["given"], ID="E")
            A = model.set_point(0, 1, classes=["given"], ID="A")
            D = model.set_point(0, -1, classes=["given"], ID="D")
            circle_ABCD = model.construct_circle(E, A)

            model.set_segment(A, E)
            model.set_segment(E, D)
            model.set_segment(A, D)
            model.set_section([A, E, D])

            # 2. Perpendicular EH to nearer chord BC
            H = model.set_point(sp.Rational(35, 76), sp.Rational(-5, 76), label="H")
            half_BC = sp.sqrt(1 - H.x**2 - H.y**2)
            len_EH = sp.sqrt(H.x**2 + H.y**2)
            ux_BC = -H.y / len_EH
            uy_BC = H.x / len_EH

            B = model.set_point(H.x + half_BC * ux_BC, H.y + half_BC * uy_BC, ID="B")
            C = model.set_point(H.x - half_BC * ux_BC, H.y - half_BC * uy_BC, ID="C")

            model.set_segment(E, H)
            model.set_segment(B, H)
            model.set_segment(H, C)
            model.set_segment(B, C)
            model.set_section([B, H, C])

            # 3. Perpendicular EK to more remote chord FG
            K = model.set_point(sp.Rational(-57, 76), sp.Rational(6, 76), label="K")
            half_FG = sp.sqrt(1 - K.x**2 - K.y**2)
            len_EK = sp.sqrt(K.x**2 + K.y**2)
            ux_FG = -K.y / len_EK
            uy_FG = K.x / len_EK

            F = model.set_point(K.x + half_FG * ux_FG, K.y + half_FG * uy_FG, ID="F")
            G = model.set_point(K.x - half_FG * ux_FG, K.y - half_FG * uy_FG, ID="G")

            # 4. Point L on EK with EL = EH, and chord MN through L perpendicular to EK
            L = model.set_point(K.x * len_EH / len_EK, K.y * len_EH / len_EK, ID="L")
            M = model.set_point(L.x + half_BC * ux_FG, L.y + half_BC * uy_FG, ID="M")
            N = model.set_point(L.x - half_BC * ux_FG, L.y - half_BC * uy_FG, ID="N")

            model.set_segment(E, L)
            model.set_segment(L, K)
            model.set_segment(E, K)
            model.set_section([E, L, K])

            model.set_segment(F, K)
            model.set_segment(K, G)
            model.set_segment(F, G)
            model.set_section([F, K, G])

            model.set_segment(M, L)
            model.set_segment(L, N)
            model.set_segment(M, N)
            model.set_section([M, L, N])

            # 5. Radii ME, EN, FE, EG and triangles MEN, FEG
            model.set_segment(M, E)
            model.set_segment(E, N)
            model.set_segment(F, E)
            model.set_segment(E, G)

            model.set_polygon([M, E, N])
            model.set_polygon([F, E, G])

            return model


        if __name__ == "__main__":
            construct_g1()