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X₉₇₀¶

CENTER OF THE APOLLONIUS CIRCLE

The Apollonius circle is described at X₁₈₁ as the circle tangent to the three excircles and encompassing them. That X₉₇₀ is its center was noted on New Year’s Day, 2002, by Paul Yiu. (Hyacinthos #4619-4623)

X₉₇₀ (CENTER OF THE APOLLONIUS CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_970 = CENTER OF THE APOLLONIUS CIRCLE."""

        from __future__ import annotations

        from pathlib import Path

        import sympy.geometry as spg
        from geometor.model import Model

        from geometor.etc.ops import (
            central_line,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            nine_point_center_of,
            promote_point_to_top,
            spieker_center_of,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00970(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_970 = CENTER OF THE APOLLONIUS CIRCLE on reference triangle ABC."""
            X_970 = eval_trilinear_point(model, A, B, C, 'a*(a**(3)*(b + c)**(2) + a*(a*b + a*c - 2*b*c)*(b**(2) + c**(2)) - b*c*(b**(3) + c**(3)) - a*(b**(4) + c**(4)) - (b**(5) + c**(5)))', ID="X_970", classes=["selected", "target"])
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            central_line(model, X_3, X_6, 3, 6, incident=True)
            X_5 = nine_point_center_of(model, A, B, C, ID="X_5", classes=["center", "incident"])
            X_10 = spieker_center_of(model, A, B, C, ID="X_10", classes=["center", "incident"])
            central_line(model, X_5, X_10, 5, 10, incident=True)
            X_21 = eval_trilinear_point(model, A, B, C, '1/(cos(B) + cos(C))', ID="X_21", classes=["center", "incident"])
            X_51 = eval_trilinear_point(model, A, B, C, 'a**(2)*cos(B - C)', ID="X_51", classes=["center", "incident"])
            central_line(model, X_21, X_51, 21, 51, incident=True)
            return promote_point_to_top(model, X_970)


        def build_model(
            sides: tuple[float, float, float] = (5.0, 6.0, 7.0),
            show_guides: bool = False,
        ) -> Model:
            """Build a `geometor.model.Model` for X_970 on the given reference triangle."""
            model = create_model('X_970 = CENTER OF THE APOLLONIUS CIRCLE')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00970(model, A, B, C)
            return model


        if __name__ == "__main__":
            here = Path(__file__).resolve().parent
            model_acute = build_model((5.0, 6.0, 7.0))
            model_acute.save(str(here / "x-00970.json"))
            write_center_svg(
                model_acute,
                here / "x-00970.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X970¶

X₉₇₀¶