G.110 line of centers of externally touching circles — Code

III.12

If two circles touch one another externally, the segment joining their centers passes through the point of contact.

"""Programmatic construction for Book III, Proposition 12 (III.12).

        If two circles touch one another externally, the straight line joining their
        centres will pass through the point of contact (reductio ad absurdum figure).
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0034.g1 using geometor.model."""
            model = Model("III.12")

            # Point of external contact A and true circle centres for the two tangent circles
            A = model.set_point(0, 0, classes=["given"], ID="A")
            O1 = model.set_point(-49, 23, classes=["auxiliary"])
            O2 = model.set_point(60, -28, classes=["auxiliary"])

            # Circle ABC and circle ADE touching externally at A
            model.construct_circle(O1, A)
            model.construct_circle(O2, A)

            B = model.set_point(-49, 23 + sp.sqrt(2930), label="B")
            E = model.set_point(60 + sp.sqrt(4384), -28, label="E")

            # Assumed centres F and G in the reductio proof
            F = model.set_point(-53, 12, classes=["given"], ID="F")
            G = model.set_point(47, -52, classes=["given"], ID="G")

            # Straight line FCDG joining the assumed centres F and G
            model.construct_line(F, G)

            # Intersections C (on circle ABC) and D (on circle ADE) along FG
            C = model.set_point(sp.Rational(-53, 5), sp.Rational(-76, 5), label="C")
            D = model.set_point(sp.Rational(-28, 5), sp.Rational(-92, 5), label="D")

            # Radii joined to the point of contact A
            model.set_segment(F, A)
            model.set_segment(G, A)

            # Collinear segments along FCDG
            model.set_segment(F, C)
            model.set_segment(C, D)
            model.set_segment(D, G)
            model.set_segment(F, D)
            model.set_segment(C, G)
            model.set_segment(F, G)
            model.set_section([F, C, G])
            model.set_section([F, D, G])

            # Triangle FAG formed in the reductio hypothesis
            model.set_polygon([F, A, G])

            return model


        if __name__ == "__main__":
            construct_g1()