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X₈₄₉¶

4th HATZIPOLAKIS-YIU POINT

Let D denote the circumcircle of triangle ABC. Let D_A be the circle tangent to sideline BC and tangent to D at A. Let B_A = AC∩D_A and C_A = AB∩D_A, and define C_B, C_B and A_C, A_B cyclically. Define A' = C_BA_B∩A_CB_C, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to ABC, and X₈₄₉ is the center of homothety. See A. Hatzipolakis and P. Yiu, Hyacinthos #2056-2070, December, 2000.

X₈₄₉ (4th HATZIPOLAKIS-YIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_849 = 4th 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 (
            central_line,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00849(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_849 = 4th HATZIPOLAKIS-YIU POINT on reference triangle ABC."""
            X_849 = eval_trilinear_point(model, A, B, C, '(a/(b + c))**(2)', ID="X_849", classes=["selected", "target"])
            X_36 = eval_trilinear_point(model, A, B, C, '1 - 2*cos(A)', ID="X_36", classes=["center", "incident"])
            X_58 = eval_trilinear_point(model, A, B, C, 'a/(b + c)', ID="X_58", classes=["center", "incident"])
            central_line(model, X_36, X_58, 36, 58, incident=True)
            X_32 = eval_trilinear_point(model, A, B, C, 'a**(3)', ID="X_32", classes=["center", "incident"])
            X_163 = eval_trilinear_point(model, A, B, C, '(sin(2*A))/(sin(2*B) - sin(2*C))', ID="X_163", classes=["center", "incident"])
            central_line(model, X_32, X_163, 32, 163, incident=True)
            X_110 = eval_trilinear_point(model, A, B, C, 'csc(B - C)', ID="X_110", classes=["center", "incident"])
            X_595 = eval_trilinear_point(model, A, B, C, 'r + R*(cos(2*A) - 1)', ID="X_595", classes=["center", "incident"])
            central_line(model, X_110, X_595, 110, 595, incident=True)
            return promote_point_to_top(model, X_849)


        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_849 on the given reference triangle."""
            model = create_model('X_849 = 4th HATZIPOLAKIS-YIU POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00849(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-00849.json"))
            write_center_svg(
                model_acute,
                here / "x-00849.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X849¶

X₈₄₉¶