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X₃₂₃₂¶

2nd TRISECTED PERIMETER POINT

There exist points A', B', C' on segments BC, CA, AB, respectively, such that B'C + C'B = C'A + A'C = A'B + B'A = (a + b + c)/3, and the lines AA', BB', CC' concur in X₃₂₃₂. Near the beginning of the 21st century, trilinears \(x : y : z\) were found for X₃₂₃₂ in terms of the unique real root of a cubic polynomial related to the cubic polynomial shown at X₃₆₉, the 1st trisected perimeter point. The proof is given in Sadi Abu-Saymeh, Mawaffaq Hajja, and Hellmuth Stachel, “Another cubic associated with the triangle,” forthcoming in Journal for Geometry and Graphics.

X₃₂₃₂ (2nd TRISECTED PERIMETER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3232 = 2nd TRISECTED PERIMETER POINT."""

        from __future__ import annotations

        from pathlib import Path

        import sympy.geometry as spg
        from geometor.model import Model

        from geometor.etc.ops import (
            construct_baseline_triangle,
            create_model,
            isotomic_conjugate,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03232(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3232 = 2nd TRISECTED PERIMETER POINT on reference triangle ABC."""
            X_369 = point_from_barycentrics(model, A, B, C, 0.35886745, 0.73098184, -0.089849297, ID="X_369", classes=["center"])
            X_3232 = isotomic_conjugate(model, A, B, C, X_369, ID="X_3232", classes=["selected", "target"])
            return promote_point_to_top(model, X_3232)


        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_3232 on the given reference triangle."""
            model = create_model('X_3232 = 2nd TRISECTED PERIMETER POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03232(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-03232.json"))
            write_center_svg(
                model_acute,
                here / "x-03232.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3232¶

X₃₂₃₂¶