G.118 perpendicular at extremity of a diameter is tangent — Code

III.16

The line drawn at right angles to the diameter of a circle from its extremity falls outside the circle, no other line can be interposed between it and the arc, and the angle of the semicircle is greater than any acute rectilineal angle.

"""Programmatic construction for Book III, Proposition 16 (III.16).

        The straight line drawn at right angles to the diameter of a circle from its
        extremity will fall outside the circle, and into the space between the straight
        line and the circumference another straight line cannot be interposed; further
        the angle of the semicircle is greater, and the remaining angle less, than any
        acute rectilineal angle.
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure for Book III, Proposition 16 using geometor.model."""
            model = Model("III.16")

            # 1. Circle ABC about centre D and diameter AB
            D = model.set_point(0, 0, classes=["given"], ID="D")
            B = model.set_point(0, 1, classes=["given"], ID="B")
            A = model.set_point(0, -1, classes=["given"], ID="A")
            circle_ABC = model.construct_circle(D, A)

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

            # 2. First reductio: suppose perpendicular from A falls within as CA; join DC
            cx = sp.Rational(-27, 35)
            cy = -sp.sqrt(1 - cx**2)
            C = model.set_point(cx, cy, ID="C")
            model.set_segment(C, A)
            model.set_segment(D, C)
            model.set_polygon([A, C, D])

            # 3. Tangent AE drawn from A at right angles to AB
            E = model.set_point(sp.Rational(-67, 35), -1, label="E")
            line_AE = model.construct_line(A, E)
            model.set_segment(A, E)

            # 4. Second reductio: suppose FA is interposed between AE and circumference CHA,
            #    and DG is drawn from D perpendicular to FA, cutting circumference CHA at H.
            F = model.set_point(sp.Rational(-65, 35), sp.Rational(-51, 70), label="F")
            line_AF = model.construct_line(A, F)

            # Point G on FA and point H where DG meets the circumference CHA
            t_G = sp.Rational(16, 65)
            G = model.set_point(F.x * t_G, -1 + (F.y + 1) * t_G, label="G")
            len_DG = sp.sqrt(G.x**2 + G.y**2)
            H = model.set_point(G.x / len_DG, G.y / len_DG, ID="H")

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

            model.set_segment(D, H)
            model.set_segment(H, G)
            model.set_segment(D, G)
            model.set_section([D, H, G])

            model.set_polygon([A, G, D])

            return model


        if __name__ == "__main__":
            construct_g1()