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X₁₁₄₁¶

GIBERT POINT

In Jean-Pierre Ehrmann and Bernard Gibert, “Special Isocubics,” downloadable from Cubics in the Triangle Plane, the point X₁₁₄₁ is labeled E, barycentrics are given, and it is established that this point also lies on the line X₅-to-X₁₁₀ [listed below as 5,49], two other cubics, and the hyperbola that passes through the points A, B, C, X₄, X₅.

X₁₁₄₁ (GIBERT POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1141 = GIBERT 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,
            nine_point_center_of,
            orthocenter_of,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01141(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1141 = GIBERT POINT on reference triangle ABC."""
            X_1141 = eval_trilinear_point(model, A, B, C, 'b*c/(16*D**(2) + (a**(2)+c**(2)-b**(2))*(a**(2)+b**(2)-c**(2)))*(16*D**(2) - 3*(b**(2)+c**(2)-a**(2))**(2))', ID="X_1141", classes=["selected", "target"])
            X_5 = nine_point_center_of(model, A, B, C, ID="X_5", classes=["center", "incident"])
            X_49 = eval_trilinear_point(model, A, B, C, 'cos(3*A)', ID="X_49", classes=["center", "incident"])
            central_line(model, X_5, X_49, 5, 49, incident=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_128 = eval_trilinear_point(model, A, B, C, '((sec(A))*(cos(2*B) + cos(2*C))*(1 + 2*cos(2*A))*(cos(2*A) + 2*cos(2*B)*cos(2*C)))', ID="X_128", classes=["center", "incident"])
            central_line(model, X_2, X_128, 2, 128, incident=True)
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            X_137 = point_from_barycentrics(model, A, B, C, 0.1354431, 0.64976082, 0.21479608, ID="X_137", classes=["center", "incident"])
            central_line(model, X_4, X_137, 4, 137, incident=True)
            return promote_point_to_top(model, X_1141)


        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_1141 on the given reference triangle."""
            model = create_model('X_1141 = GIBERT POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01141(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-01141.json"))
            write_center_svg(
                model_acute,
                here / "x-01141.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1141¶

X₁₁₄₁¶