G.25 base angles of an isosceles triangle — Code

I.5

In an isosceles triangle, the angles at the base are equal, and if the equal sides are extended, the angles under the base are equal.
A B C F G D E 
"""Programmatic construction for Book I, Proposition 5 (I.5).

        In isosceles triangles the angles at the base are equal to one another,
        and, if the equal straight lines be produced further, the angles under the
        base will be equal to one another (Pons Asinorum).
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure for Book I, Proposition 5 using geometor.model."""
            model = Model("I.5")

            # 1. Let ABC be an isosceles triangle having the side AB equal to the side AC.
            A = model.set_point(0, sp.Rational(43, 19), classes=["given"], label="A")
            B = model.set_point(-1, 0, classes=["given"], ID="B")
            C = model.set_point(1, 0, classes=["given"], ID="C")

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

            # 2. Let the straight lines BD, CE be produced further in a straight line with AB, AC [Post. 2].
            line_AB = model.construct_line(A, B)
            line_AC = model.construct_line(A, C)

            # Outer endpoints D and E on the produced rays
            t_D = sp.Rational(153, 86)
            D = model.set_point(-t_D, sp.Rational(43, 19) * (1 - t_D), label="D")
            E = model.set_point(t_D, sp.Rational(43, 19) * (1 - t_D), label="E")

            # 3. Let a point F be taken at random on BD;
            #    from AE the greater let AG be cut off equal to AF the less [I. 3].
            t_F = sp.Rational(125, 86)
            F = model.set_point(-t_F, sp.Rational(43, 19) * (1 - t_F), label="F")
            G = model.set_point(t_F, sp.Rational(43, 19) * (1 - t_F), label="G")

            # Collinear sections and sub-segments along left ray ABD
            model.set_section([A, B, F])
            model.set_section([B, F, D])
            model.set_segment(B, F)
            model.set_segment(F, D)
            model.set_segment(A, F)
            model.set_segment(B, D)
            model.set_segment(A, D)

            # Collinear sections and sub-segments along right ray ACE
            model.set_section([A, C, G])
            model.set_section([C, G, E])
            model.set_segment(C, G)
            model.set_segment(G, E)
            model.set_segment(A, G)
            model.set_segment(C, E)
            model.set_segment(A, E)

            # 4. And let the straight lines FC, GB be joined [Post. 1].
            model.set_segment(F, C)
            model.set_segment(G, B)

            # 5. Triangles compared in the proof: AFC, AGB and BFC, CGB
            model.set_polygon([A, F, C])
            model.set_polygon([A, G, B])
            model.set_polygon([B, F, C])
            model.set_polygon([C, G, B])

            return model


        if __name__ == "__main__":
            construct_g1()