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X₃₄₁₁¶

1st BROCARD-KIEPERT-FERMAT CUSP

X₃₄₁₁ and X₃₄₁₂ occur in Bernard Gibert’s study of the Brocard-Kiepert quartic, which has three cusps: X₂₀ with Euler line as tangent; X₃₄₁₁ with line X₅ X₁₃ as tangent, and X₃₄₁₂ with line X₅ X₁₄ as tangent.

X₃₄₁₁ (1st BROCARD-KIEPERT-FERMAT CUSP) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3411 = 1st BROCARD-KIEPERT-FERMAT CUSP."""

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


        def construct_x_03411(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3411 = 1st BROCARD-KIEPERT-FERMAT CUSP on reference triangle ABC."""
            X_3411 = eval_trilinear_point(model, A, B, C, '(cos(A) - 2*sin(B)*sin(C) - 3*cos(A + pi  /3))', ID="X_3411", classes=["selected", "target"])
            X_5 = nine_point_center_of(model, A, B, C, ID="X_5", classes=["center", "incident"])
            X_13 = eval_trilinear_point(model, A, B, C, 'csc(A +  pi  /3)', ID="X_13", classes=["center", "incident"])
            central_line(model, X_5, X_13, 5, 13, incident=True)
            X_16 = eval_trilinear_point(model, A, B, C, 'sin(A-  pi  /3)', ID="X_16", classes=["center", "incident"])
            X_20 = eval_trilinear_point(model, A, B, C, 'cos(A) - cos(B)*cos(C)', ID="X_20", classes=["center", "incident"])
            central_line(model, X_16, X_20, 16, 20, incident=True)
            X_14 = eval_trilinear_point(model, A, B, C, 'csc(A -  pi  /3)', ID="X_14", classes=["center", "incident"])
            X_382 = eval_trilinear_point(model, A, B, C, 'cos(A) - 4*cos(B)*cos(C)', ID="X_382", classes=["center", "incident"])
            central_line(model, X_14, X_382, 14, 382, incident=True)
            return promote_point_to_top(model, X_3411)


        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_3411 on the given reference triangle."""
            model = create_model('X_3411 = 1st BROCARD-KIEPERT-FERMAT CUSP')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03411(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-03411.json"))
            write_center_svg(
                model_acute,
                here / "x-03411.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3411¶

X₃₄₁₁¶