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X₃₆₈¶

EQUI-BROCARD CENTER

The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Peter Yff proved that X₃₆₈ lies on the self-isogonal conjugate cubic with trilinear equation f(a,b,c)u + f(b,c,a)v + f(c,a,b)w = 0, where \(f(a,b,c) = bc(b^{2} - c^{2})\) and, for variable \(x : y : z\), the cubics u, v, w are given by u(x,y,z) = \(x(y^{2}\) + \(z^{2})\), v = u(y,z,x), w = u(z,x,y).

X₃₆₈ (EQUI-BROCARD CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_368 = EQUI-BROCARD CENTER."""

        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_00368(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_368 = EQUI-BROCARD CENTER on reference triangle ABC."""
            X_368 = point_from_barycentrics(model, A, B, C, 0.3279321, 0.34259259, 0.32947531, ID="X_368", classes=["selected", "target"])
            return promote_point_to_top(model, X_368)


        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_368 on the given reference triangle."""
            model = create_model('X_368 = EQUI-BROCARD CENTER')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00368(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-00368.json"))
            write_center_svg(
                model_acute,
                here / "x-00368.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X368¶

X₃₆₈¶