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X₆₀₀¶

3rd HATZIPOLAKIS-YIU POINT

Let O_A be the circle with center A and radius R, the circumradius of triangle ABC. Let B_A be the point where O_A meets line AB farthest from B. Define C_B and A_C cyclically. Let C_A be the point where O_A meets line AC farthest from C. Define A_B and B_C cyclically. X₆₀₀ is the radical center of the circles AB_AC_A, BC_BA_B, CA_CB_C. If “farthest from” is changed to “nearest to” in the construction, the resulting point is X₅₅₀₇. (Antreas Hatzipolakis, Paul Yiu, 1/5/01, Hyacinthos #2344, #2346-8; for a correction in coordinates, see Paul Yiu, Adgeom #202.)

X₆₀₀ (3rd HATZIPOLAKIS-YIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_600 = 3rd HATZIPOLAKIS-YIU POINT."""

        from __future__ import annotations

        from pathlib import Path

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

        from geometor.etc.ops import (
            construct_baseline_triangle,
            create_model,
            eval_barycentric_point,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00600(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_600 = 3rd HATZIPOLAKIS-YIU POINT on reference triangle ABC."""
            X_600 = eval_barycentric_point(model, A, B, C, '(a**(2)*(b*c + 2*S)*(a*b*c*(b + c - a) + 2*(b**(2) + c**(2) - a**(2))*S))', ID="X_600", classes=["selected", "target"])
            return promote_point_to_top(model, X_600)


        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_600 on the given reference triangle."""
            model = create_model('X_600 = 3rd HATZIPOLAKIS-YIU POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00600(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-00600.json"))
            write_center_svg(
                model_acute,
                here / "x-00600.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X600¶

X₆₀₀¶