G.121 perpendicular to tangent at contact passes through center — Code

III.19

If a line touches a circle, the line drawn at right angles to the tangent from the point of contact passes through the center of the circle.

"""Programmatic construction for Book III, Proposition 19 (III.19).

        If a straight line touch a circle, and from the point of contact a straight
        line be drawn at right angles to the tangent, the centre of the circle will be
        on the straight line so drawn.
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure for Book III, Proposition 19 using geometor.model."""
            model = Model("III.19")

            # 1. Circle ABC touched by straight line DE at C, and CA drawn at right angles to DE.
            O = model.set_point(0, 0, ID="O")
            C = model.set_point(0, -1, classes=["given"], ID="C")
            A = model.set_point(0, 1, classes=["given"], ID="A")
            B = model.set_point(sp.Rational(60, 61), sp.Rational(11, 61), label="B")

            circle_ABC = model.construct_circle(O, C)

            D = model.set_point(sp.Rational(-43, 40), -1, label="D")
            E = model.set_point(sp.Rational(109, 80), -1, label="E")

            line_DE = model.construct_line(D, E)
            model.set_segment(D, C)
            model.set_segment(C, E)
            model.set_segment(D, E)
            model.set_section([D, C, E])

            model.set_segment(C, A)

            # 2. Reductio: suppose F is the centre, and join CF.
            F = model.set_point(sp.Rational(13, 40), sp.Rational(3, 40), label="F")
            model.set_segment(C, F)

            return model


        if __name__ == "__main__":
            construct_g1()