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X₁₁₃₁¶

ARCTAN(2) KIEPERT POINT

To construct the Vecten point, X₄₈₅, squares are erected outward on the sides of ABC. If A', B', C' are the centers of these squares, then triangle A'B'C' is perspective to ABC with perspector X₄₈₅. Now let A″ be the midpoint of the side of the A-square that does not touch line BC, and define B″ and C″ cyclically. Then triangle A″B″C″ is perspective to ABC with perspector X₁₁₃₁. Angle(A″BC) = angle(B″CA) = angle(C″AB), so that X₁₁₃₁ lies on the Kiepert hyperbola. Here, the common angle is arctan(2). (Darij Grinberg 9/22/02)

X₁₁₃₁ (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_1131 = 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_01131(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1131 = ARCTAN(2) KIEPERT POINT on reference triangle ABC."""
            X_1131 = eval_trilinear_point(model, A, B, C, '1/(sin(A) + 2*cos(A))', ID="X_1131", classes=["selected", "target"])
            X_175 = eval_trilinear_point(model, A, B, C, '-1 + sec(A/2)*cos(B/2)*cos(C/2)', ID="X_175", 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_175, X_226, 175, 226, incident=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_490 = eval_trilinear_point(model, A, B, C, '((cos(A))*(1 + csc(A)*sin(B)*sin(C)) - cos(B)*cos(C))', ID="X_490", classes=["center", "incident"])
            central_line(model, X_2, X_490, 2, 490, incident=True)
            X_20 = eval_trilinear_point(model, A, B, C, 'cos(A) - cos(B)*cos(C)', ID="X_20", classes=["center", "incident"])
            X_485 = eval_trilinear_point(model, A, B, C, 'sec(A -  pi  /4)', ID="X_485", classes=["center", "incident"])
            central_line(model, X_20, X_485, 20, 485, incident=True)
            return promote_point_to_top(model, X_1131)


        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_1131 on the given reference triangle."""
            model = create_model('X_1131 = ARCTAN(2) KIEPERT POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01131(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-01131.json"))
            write_center_svg(
                model_acute,
                here / "x-01131.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1131¶

X₁₁₃₁¶