G.107 point with three equal segments to a circle is the center — Code

III.9

A point within a circle from which more than two equal segments fall on the circle is the center of the circle.

"""Programmatic construction for Book III, Proposition 9 (III.9).

        If a point D be taken within a circle ABC, and more than two equal straight
        lines DA, DB, DC fall from D on the circle, the point D is the centre of the
        circle ABC (proved via perpendicular bisectors GK of AB and HL of BC).
        """

        from __future__ import annotations

        import sympy as sp
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0027.g1 for Book III, Proposition 9."""
            model = Model("III.9")

            # Circle ABC with centre D at origin and horizontal diameter KG
            D = model.set_point(0, 0, classes=["given"], ID="D")
            G = model.set_point(1, 0, ID="G")
            K = model.set_point(-1, 0, ID="K")

            model.construct_line(K, G)
            model.construct_circle(D, G)

            # Vertical chord AB bisected perpendicularly at E(-3/4, 0) by KG
            E = model.set_point(sp.Rational(-3, 4), 0, label="E")
            B = model.set_point(sp.Rational(-3, 4), sp.sqrt(7) / 4, classes=["given"], label="B")
            A = model.set_point(sp.Rational(-3, 4), -sp.sqrt(7) / 4, classes=["given"], label="A")
            model.construct_line(A, B)

            # Point C on circumference just above G
            C = model.set_point(sp.Rational(199, 200), sp.sqrt(399) / 200, classes=["given"], label="C")
            model.construct_line(B, C)

            # Midpoint F of BC and perpendicular bisector diameter HL through D and F
            F = model.set_point((B.x + C.x) / 2, (B.y + C.y) / 2, label="F")
            model.construct_line(D, F)
            len_DF = sp.sqrt(F.x**2 + F.y**2)
            L = model.set_point(F.x / len_DF, F.y / len_DF, ID="L")
            H = model.set_point(-F.x / len_DF, -F.y / len_DF, ID="H")

            # Equal straight lines DA, DB, DC
            model.set_segment(D, A)
            model.set_segment(D, B)
            model.set_segment(D, C)

            # Chord AB and sub-segments AE, EB
            model.set_segment(A, E)
            model.set_segment(E, B)
            model.set_segment(A, B)
            model.set_section([A, E, B])

            # Chord BC and sub-segments BF, FC
            model.set_segment(B, F)
            model.set_segment(F, C)
            model.set_segment(B, C)
            model.set_section([B, F, C])

            # Diameter KG and sub-segments KE, ED, DG
            model.set_segment(K, E)
            model.set_segment(E, D)
            model.set_segment(D, G)
            model.set_segment(K, D)
            model.set_segment(E, G)
            model.set_segment(K, G)

            # Diameter HL and sub-segments HD, DF, FL
            model.set_segment(H, D)
            model.set_segment(D, F)
            model.set_segment(F, L)
            model.set_segment(H, F)
            model.set_segment(D, L)
            model.set_segment(H, L)

            # Triangles AED and BED
            model.set_polygon([A, E, D])
            model.set_polygon([B, E, D])

            return model


        if __name__ == "__main__":
            construct_g1()