← etc / Encyclopedia of Triangle Centers

X₂₇₇₈¶

ISOGONAL CONJUGATE OF X₂₆₉₄

As the isogonal conjugate of a point on the circumcircle, X₂₇₇₈ lies on the line at infinity.

X₂₇₇₈ (ISOGONAL CONJUGATE OF X₂₆₉₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_2778 = ISOGONAL CONJUGATE OF X_2694."""

        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,
            isogonal_conjugate,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_02778(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_2778 = ISOGONAL CONJUGATE OF X_2694 on reference triangle ABC."""
            X_2694 = point_from_barycentrics(model, A, B, C, 1.0089528, 0.48910827, -0.49806112, ID="X_2694", classes=["center"])
            X_2778 = isogonal_conjugate(model, A, B, C, X_2694, ID="X_2778", classes=["selected", "target"])
            X_65 = eval_trilinear_point(model, A, B, C, 'cos(B) + cos(C)', ID="X_65", classes=["center", "incident"])
            X_74 = eval_trilinear_point(model, A, B, C, '1/(cos(A) - 2*cos(B)*cos(C))', ID="X_74", classes=["center", "incident"])
            central_line(model, X_65, X_74, 65, 74, 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_123 = eval_trilinear_point(model, A, B, C, '(csc(A))*(sec(B) - sec(C))*((sec(A))*((sin(B))**(2) - (sin(C))**(2)) + sin(C)*tan(C) - sin(B)*tan(B))', ID="X_123", classes=["center", "incident"])
            central_line(model, X_113, X_123, 113, 123, incident=True)
            X_30 = eval_trilinear_point(model, A, B, C, 'cos(A) - 2*cos(B)*cos(C)', ID="X_30", classes=["center", "incident"])
            X_511 = eval_trilinear_point(model, A, B, C, 'cos(A +  omega  )', ID="X_511", classes=["center", "incident"])
            central_line(model, X_30, X_511, 30, 511, incident=True)
            return promote_point_to_top(model, X_2778)


        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_2778 on the given reference triangle."""
            model = create_model('X_2778 = ISOGONAL CONJUGATE OF X_2694')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_02778(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-02778.json"))
            write_center_svg(
                model_acute,
                here / "x-02778.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X2778¶

X₂₇₇₈¶