← etc / Encyclopedia of Triangle Centers

X₆₉₀¶

CROSSDIFFERENCE OF LINE X₆ AND X₁₁₀

As the isogonal conjugate of a point on the circumcircle, X₆₉₀ lies on the line at infinity.

X₆₉₀ (CROSSDIFFERENCE OF LINE X₆ AND X₁₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_690 = CROSSDIFFERENCE OF LINE X_6 AND X_110."""

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


        def construct_x_00690(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_690 = CROSSDIFFERENCE OF LINE X_6 AND X_110 on reference triangle ABC."""
            X_690 = eval_trilinear_point(model, A, B, C, 'b*c*(b**(2) - c**(2))*(2*a**(2) - b**(2) - c**(2))', ID="X_690", classes=["selected", "target"])
            X_74 = eval_trilinear_point(model, A, B, C, '1/(cos(A) - 2*cos(B)*cos(C))', ID="X_74", classes=["center", "incident"])
            X_98 = eval_trilinear_point(model, A, B, C, 'sec(A +  omega  )', ID="X_98", classes=["center", "incident"])
            central_line(model, X_74, X_98, 74, 98, incident=True)
            X_99 = eval_trilinear_point(model, A, B, C, 'b*c/(b**(2) - c**(2))', ID="X_99", classes=["center", "incident"])
            X_110 = eval_trilinear_point(model, A, B, C, 'csc(B - C)', ID="X_110", classes=["center", "incident"])
            central_line(model, X_99, X_110, 99, 110, incident=True)
            X_113 = eval_trilinear_point(model, A, B, C, 'sin(B)*sin(C)*((sin(C))/(cos(C) - 2*cos(A)*cos(B)) + (sin(B))/(cos(B) - 2*cos(A)*cos(C)))', ID="X_113", classes=["center", "incident"])
            X_114 = eval_trilinear_point(model, A, B, C, 'b*c*(b*sec(B +  omega  ) + c*sec(C +  omega  ))', ID="X_114", classes=["center", "incident"])
            central_line(model, X_113, X_114, 113, 114, incident=True)
            return promote_point_to_top(model, X_690)


        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_690 on the given reference triangle."""
            model = create_model('X_690 = CROSSDIFFERENCE OF LINE X_6 AND X_110')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00690(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-00690.json"))
            write_center_svg(
                model_acute,
                here / "x-00690.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X690¶

X₆₉₀¶