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X₁₁₁₈¶

1st HATZIPOLAKIS PERSPECTOR

Let T_aT_bT_c be the intouch triangle of ABC. Let C_A be the point other than T_c in which the perpendicular to BC from T_c meets the incircle, let B_A be the point other than T_b in which the perpendicular to BC from T_b meets the incircle, and let A₀ be the point of intersection of lines BC_A and CB_A. Define B₀ and C₀ cyclically. Then triangle A₀B₀C₀ is perspective to ABC, and the perspector is X₁₁₁₈. (Antreas Hatzipolakis, #5321, 4/30/02)

X₁₁₁₈ (1st HATZIPOLAKIS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1118 = 1st HATZIPOLAKIS PERSPECTOR."""

        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,
            orthocenter_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01118(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1118 = 1st HATZIPOLAKIS PERSPECTOR on reference triangle ABC."""
            X_1118 = eval_trilinear_point(model, A, B, C, '((1 - cos(A))/(cos(A))**(2))', ID="X_1118", classes=["selected", "target"])
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            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_28 = eval_trilinear_point(model, A, B, C, '(tan(A))/(b + c)', ID="X_28", classes=["center", "incident"])
            X_56 = eval_trilinear_point(model, A, B, C, 'a/(b + c - a)', ID="X_56", classes=["center", "incident"])
            central_line(model, X_28, X_56, 28, 56, incident=True)
            X_24 = eval_trilinear_point(model, A, B, C, 'sec(A)*cos(2*A)', ID="X_24", classes=["center", "incident"])
            X_108 = eval_trilinear_point(model, A, B, C, 'a/(sec(B) - sec(C))', ID="X_108", classes=["center", "incident"])
            central_line(model, X_24, X_108, 24, 108, incident=True)
            return promote_point_to_top(model, X_1118)


        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_1118 on the given reference triangle."""
            model = create_model('X_1118 = 1st HATZIPOLAKIS PERSPECTOR')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01118(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-01118.json"))
            write_center_svg(
                model_acute,
                here / "x-01118.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1118¶

X₁₁₁₈¶