G.109 line of centers of internally touching circles — Code

III.11

If two circles touch one another internally, the line joining their centers passes through the point of contact.
A B C D E F G H 
"""Programmatic construction for Book III, Proposition 11 (III.11).

        If two circles ABC, ADE touch one another internally at A, and their centres F
        and G be taken, the straight line joining their centres and produced will fall
        on the point of contact A (reductio ad absurdum via line FGH).
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0030.g2 for Book III, Proposition 11."""
            model = Model("III.11")

            # Outer circle ABC with centre F at origin and radius 1
            F = model.set_point(0, 0, classes=["given"], ID="F")
            H = model.set_point(0, 1, ID="H")
            B = model.set_point(1, 0, ID="B")
            C = model.set_point(sp.Rational(-1, 3), -sp.Rational(2, 3) * sp.sqrt(2), label="C")
            model.construct_circle(F, H)

            # Point of internal contact A on the upper-left circumference
            A = model.set_point(sp.Rational(-4, 5), sp.Rational(3, 5), classes=["given"], label="A")

            # Inner circle ADE touching ABC internally at A, with radius 5/7
            r_in = sp.Rational(5, 7)
            C_in = model.set_point((1 - r_in) * A.x, (1 - r_in) * A.y)
            model.construct_circle(C_in, A)

            # Hypothetical centre G on vertical line FH, and intersection D of FH with ADE
            G = model.set_point(0, sp.Rational(16, 49), classes=["given"], label="G")
            model.construct_line(F, H)
            y_D = C_in.y + sp.sqrt(r_in**2 - C_in.x**2)
            D = model.set_point(0, y_D, ID="D")
            E = model.set_point(C_in.x, C_in.y - r_in, ID="E")

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

            # Radii AF and AG, and triangle AFG
            model.set_segment(A, F)
            model.set_segment(A, G)
            model.set_polygon([A, F, G])

            return model


        if __name__ == "__main__":
            construct_g1()