← etc / Encyclopedia of Triangle Centers

X₅₂₂¶

ISOGONAL CONJUGATE OF X₁₀₉

As the isogonal conjugate of a point on the circumcircle, X₅₂₂ lies on the line at infinity.

X₅₂₂ (ISOGONAL CONJUGATE OF X₁₀₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_522 = ISOGONAL CONJUGATE OF X_109."""

        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,
            mittenpunkt_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00522(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_522 = ISOGONAL CONJUGATE OF X_109 on reference triangle ABC."""
            X_109 = eval_trilinear_point(model, A, B, C, 'a/(cos(B) - cos(C))', ID="X_109", classes=["center"])
            X_522 = isogonal_conjugate(model, A, B, C, X_109, ID="X_522", classes=["selected", "target"])
            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)
            X_9 = mittenpunkt_of(model, A, B, C, ID="X_9", classes=["center", "incident"])
            X_657 = eval_trilinear_point(model, A, B, C, '(1 + cos(A))*(cos(B) - cos(C))', ID="X_657", classes=["center", "incident"])
            central_line(model, X_9, X_657, 9, 657, incident=True)
            X_100 = eval_trilinear_point(model, A, B, C, '1/(b - c)', ID="X_100", classes=["center", "incident"])
            X_655 = eval_trilinear_point(model, A, B, C, '1/(cos(A - B) - cos(C- A))', ID="X_655", classes=["center", "incident"])
            central_line(model, X_100, X_655, 100, 655, incident=True)
            return promote_point_to_top(model, X_522)


        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_522 on the given reference triangle."""
            model = create_model('X_522 = ISOGONAL CONJUGATE OF X_109')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00522(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-00522.json"))
            write_center_svg(
                model_acute,
                here / "x-00522.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X522¶

X₅₂₂¶