← etc / Encyclopedia of Triangle Centers

X₆₆₂¶

TRILINEAR POLE OF LINE X₁X_21

Let A₂B₂C₂ and A₃B₃C₃ be the 2nd and 3rd Parry triangles. Let A' be the trilinear product A₂*A₃, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₆₆₂. (Randy Hutson, February 10, 2016)

X₆₆₂ (TRILINEAR POLE OF LINE X₁X_21) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_662 = TRILINEAR POLE OF LINE X_1X_21."""

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


        def construct_x_00662(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_662 = TRILINEAR POLE OF LINE X_1X_21 on reference triangle ABC."""
            X_661 = eval_trilinear_point(model, A, B, C, 'cot(B) - cot(C)', ID="X_661", classes=["center"])
            X_662 = isogonal_conjugate(model, A, B, C, X_661, ID="X_662", classes=["selected", "target"])
            X_48 = eval_trilinear_point(model, A, B, C, 'tan(B) + tan(C)', ID="X_48", classes=["center", "incident"])
            X_75 = eval_trilinear_point(model, A, B, C, '1/a**(2)', ID="X_75", classes=["center", "incident"])
            central_line(model, X_48, X_75, 48, 75, incident=True)
            X_81 = eval_trilinear_point(model, A, B, C, '1/(b + c)', ID="X_81", classes=["center", "incident"])
            X_88 = eval_trilinear_point(model, A, B, C, '1/(b + c - 2*a)', ID="X_88", classes=["center", "incident"])
            central_line(model, X_81, X_88, 81, 88, incident=True)
            X_99 = eval_trilinear_point(model, A, B, C, 'b*c/(b**(2) - c**(2))', ID="X_99", classes=["center", "incident"])
            X_101 = eval_trilinear_point(model, A, B, C, 'a/(b - c)', ID="X_101", classes=["center", "incident"])
            central_line(model, X_99, X_101, 99, 101, incident=True)
            return promote_point_to_top(model, X_662)


        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_662 on the given reference triangle."""
            model = create_model('X_662 = TRILINEAR POLE OF LINE X_1X_21')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00662(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-00662.json"))
            write_center_svg(
                model_acute,
                here / "x-00662.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X662¶

X₆₆₂¶