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X₁₁₃₂¶

ARCTAN(-2) KIEPERT POINT

X₁₁₃₂ is constructed in the manner for X₁₁₃₁, using squares erected inward, so that the three equal angles have common measure arctan(-2), and X₁₁₃₂ lies on the Kiepert hyperbola. (Darij Grinberg 9/22/02)

X₁₁₃₂ (ARCTAN(-2) KIEPERT POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1132 = ARCTAN(-2) KIEPERT 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_01132(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1132 = ARCTAN(-2) KIEPERT POINT on reference triangle ABC."""
            X_1132 = eval_trilinear_point(model, A, B, C, '1/(sin(A) - 2*cos(A))', ID="X_1132", classes=["selected", "target"])
            X_176 = eval_trilinear_point(model, A, B, C, '1 + sec(A/2)*cos(B/2)*cos(C/2)', ID="X_176", classes=["center", "incident"])
            X_226 = eval_trilinear_point(model, A, B, C, '(csc(A))*(cos(B) + cos(C))', ID="X_226", classes=["center", "incident"])
            central_line(model, X_176, X_226, 176, 226, incident=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_489 = eval_trilinear_point(model, A, B, C, '((cos(A))*(1 - csc(A)*sin(B)*sin(C)) - cos(B)*cos(C))', ID="X_489", classes=["center", "incident"])
            central_line(model, X_2, X_489, 2, 489, incident=True)
            X_20 = eval_trilinear_point(model, A, B, C, 'cos(A) - cos(B)*cos(C)', ID="X_20", classes=["center", "incident"])
            X_486 = eval_trilinear_point(model, A, B, C, 'sec(A +  pi  /4)', ID="X_486", classes=["center", "incident"])
            central_line(model, X_20, X_486, 20, 486, incident=True)
            return promote_point_to_top(model, X_1132)


        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_1132 on the given reference triangle."""
            model = create_model('X_1132 = ARCTAN(-2) KIEPERT POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01132(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-01132.json"))
            write_center_svg(
                model_acute,
                here / "x-01132.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1132¶

X₁₁₃₂¶