G.123 inscribed angle theorem — Code

III.20

In a circle, the angle at the center is double the angle at the circumference when the angles have the same arc as base.
A B C D E F G 
"""Programmatic construction for Book III, Proposition 20 (III.20).

        In a circle the angle at the centre is double of the angle at the circumference,
        when the angles have the same circumference as base.
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure for Book III, Proposition 20 using geometor.model."""
            model = Model("III.20")

            # 1. Circle ABC about centre E.
            E = model.set_point(0, 0, classes=["given"], ID="E")
            C = model.set_point(1, 0, classes=["given"], ID="C")
            circle_ABC = model.construct_circle(E, C)

            # Points A, B, D on circumference and antipodal points F, G
            A = model.set_point(sp.Rational(-60, 61), sp.Rational(11, 61), classes=["given"], label="A")
            F = model.set_point(sp.Rational(60, 61), sp.Rational(-11, 61), label="F")
            B = model.set_point(sp.Rational(28, 45), sp.Rational(-35, 45), classes=["given"], label="B")
            D = model.set_point(sp.Rational(13, 37), sp.Rational(35, 37), label="D")
            G = model.set_point(sp.Rational(-13, 37), sp.Rational(-35, 37), label="G")

            # 2. First case: angle BEC at centre, angle BAC at circumference, diameter AEF.
            model.set_segment(A, B)
            model.set_segment(A, C)
            model.set_segment(E, B)
            model.set_segment(E, C)

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

            model.set_polygon([E, A, B])
            model.set_polygon([E, A, C])

            # 3. Second case: inflected angle BDC at circumference, diameter DEG.
            model.set_segment(D, B)
            model.set_segment(D, C)

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

            model.set_polygon([E, D, B])
            model.set_polygon([E, D, C])

            return model


        if __name__ == "__main__":
            construct_g1()