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X₃₆₉¶

1st TRISECTED PERIMETER POINT

There exist points A', B', C' on segments BC, CA, AB, respectively, such that AB' + AC' = BC' + BA' = CA' + CB' = (a + b + c)/3, and the lines AA', BB', CC' concur in X₃₆₉. Near the end of the 20th century, Yff found trilinears for X₃₆₉ in terms of the unique real root, r, of the cubic polynomial

X₃₆₉ (1st TRISECTED PERIMETER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_369 = 1st 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,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00369(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_369 = 1st 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=["selected", "target"])
            return promote_point_to_top(model, X_369)


        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_369 on the given reference triangle."""
            model = create_model('X_369 = 1st TRISECTED PERIMETER POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00369(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-00369.json"))
            write_center_svg(
                model_acute,
                here / "x-00369.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X369¶

X₃₆₉¶