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X₃₁₀₂¶

INNER VERTEX OF THE BROCARD SQUARE

The 1st Brocard point (trilinears \(c/b : a/c : b/a)\) and 2nd Brocard point (\(b/c : c/a : a/b)\) are opposing vertices of a square here called the Brocard square. The other two vertices are X₃₁₀₂ and X₃₁₀₃. Both are on the Brocard axis X₃ X₆, and X₃₁₀₂ is the closer to X₃.

X₃₁₀₂ (INNER VERTEX OF THE BROCARD SQUARE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3102 = INNER VERTEX OF THE BROCARD SQUARE."""

        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,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03102(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3102 = INNER VERTEX OF THE BROCARD SQUARE on reference triangle ABC."""
            X_3102 = eval_trilinear_point(model, A, B, C, 'cos(A +  omega  -  pi  /4)', ID="X_3102", classes=["selected", "target"])
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            central_line(model, X_3, X_6, 3, 6, incident=True)
            X_76 = eval_trilinear_point(model, A, B, C, '1/a**(3)', ID="X_76", classes=["center", "incident"])
            X_486 = eval_trilinear_point(model, A, B, C, 'sec(A +  pi  /4)', ID="X_486", classes=["center", "incident"])
            central_line(model, X_76, X_486, 76, 486, incident=True)
            X_194 = eval_trilinear_point(model, A, B, C, 'b*c*(a**(2)*b**(2) + a**(2)*c**(2) - b**(2)*c**(2))', ID="X_194", classes=["center", "incident"])
            X_488 = eval_trilinear_point(model, A, B, C, '((cos(A))*(1 + csc(A)*sin(B)*sin(C)))', ID="X_488", classes=["center", "incident"])
            central_line(model, X_194, X_488, 194, 488, incident=True)
            return promote_point_to_top(model, X_3102)


        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_3102 on the given reference triangle."""
            model = create_model('X_3102 = INNER VERTEX OF THE BROCARD SQUARE')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03102(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-03102.json"))
            write_center_svg(
                model_acute,
                here / "x-03102.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3102¶

X₃₁₀₂¶