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X₁₁₄₈¶

YIU-HATZIPOLAKIS POINT

Suppose L_A, L_B, L_C are lines through a point P, respectively perpendicular to sidelines BC, CA, AB. Let A_B be the point where L_A meets AB, and let A_C be the point where L_A meets AC. Define B_C, B_A, C_A, C_B cyclically. Then X₁₁₄₈ is the point P, which satisfies

X₁₁₄₈ (YIU-HATZIPOLAKIS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1148 = YIU-HATZIPOLAKIS 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,
            ceva_conjugate,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            incenter_of,
            orthocenter_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01148(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1148 = YIU-HATZIPOLAKIS POINT on reference triangle ABC."""
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center"])
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center"])
            X_1148 = ceva_conjugate(model, A, B, C, X_1, X_4, ID="X_1148", classes=["selected", "target"])
            X_65 = eval_trilinear_point(model, A, B, C, 'cos(B) + cos(C)', ID="X_65", classes=["center", "incident"])
            central_line(model, X_4, X_65, 4, 65, incident=True)
            X_46 = eval_trilinear_point(model, A, B, C, 'cos(B) + cos(C) - cos(A)', ID="X_46", classes=["center", "incident"])
            X_243 = eval_trilinear_point(model, A, B, C, '(sec(A))*((cos(A))**(2) - cos(B)*cos(C))', ID="X_243", classes=["center", "incident"])
            central_line(model, X_46, X_243, 46, 243, incident=True)
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_653 = eval_trilinear_point(model, A, B, C, '1/(sec(B) - sec(C))', ID="X_653", classes=["center", "incident"])
            central_line(model, X_3, X_653, 3, 653, incident=True)
            return promote_point_to_top(model, X_1148)


        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_1148 on the given reference triangle."""
            model = create_model('X_1148 = YIU-HATZIPOLAKIS POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01148(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-01148.json"))
            write_center_svg(
                model_acute,
                here / "x-01148.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1148¶

X₁₁₄₈¶