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X₅₉₃¶

1st HATZIPOLAKIS-YIU POINT

Let O(A) be the circle tangent to line BC and to the circumcircle of triangle ABC at vertex A. Let A_B and A_C be where O(A) meets lines AB and AC, respectively. Let L(A) be the line joining A_B and A_C, and define L(B) and L(C) cyclically. Let A' be where L(B) and L(C) meet, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to triangle ABC, and the center of homothety is X₅₉₃. See Antreas Hatzipolakis, Paul Yiu, Hyacinthos #2070.

X₅₉₃ (1st HATZIPOLAKIS-YIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_593 = 1st 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,
            centroid_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00593(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_593 = 1st HATZIPOLAKIS-YIU POINT on reference triangle ABC."""
            X_593 = eval_trilinear_point(model, A, B, C, 'a/(b + c)**(2)', ID="X_593", 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_31 = eval_trilinear_point(model, A, B, C, 'a**(2)', ID="X_31", classes=["center", "incident"])
            X_110 = eval_trilinear_point(model, A, B, C, 'csc(B - C)', ID="X_110", classes=["center", "incident"])
            central_line(model, X_31, X_110, 31, 110, incident=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_261 = eval_trilinear_point(model, A, B, C, '(((csc(A))*(sec(B/2 - C/2)))**(2))', ID="X_261", classes=["center", "incident"])
            central_line(model, X_2, X_261, 2, 261, incident=True)
            return promote_point_to_top(model, X_593)


        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_593 on the given reference triangle."""
            model = create_model('X_593 = 1st HATZIPOLAKIS-YIU POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00593(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-00593.json"))
            write_center_svg(
                model_acute,
                here / "x-00593.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X593¶

X₅₉₃¶