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X₃₆₁₄¶

1st HUTSON-FEUERBACH POINT

Let Eu_aEu_bEu_c be the Feuerbach triangle, L the line through A and X₅, and A″ = L∩Eu_bEu_c; define B″ and C″ cyclically. Then the lines Eu_aA″, Eu_bB″, Eu_cC″ concur in X₃₆₁₄. X₃₆₁₄ = {X₅,X₁₂}-harmonic conjugate of X₁₁. Further, X₃₆₁₄ is the trilinear pole (with respect to the Feuerbach triangle) of the perspectrix of ABC and the Feuerbach triangle. (Randy Hutson, August 30, 2011.)

X₃₆₁₄ (1st HUTSON-FEUERBACH POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3614 = 1st HUTSON-FEUERBACH POINT."""

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


        def construct_x_03614(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3614 = 1st HUTSON-FEUERBACH POINT on reference triangle ABC."""
            X_3614 = eval_trilinear_point(model, A, B, C, '1 + 3*cos(B - C)', ID="X_3614", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_5 = nine_point_center_of(model, A, B, C, ID="X_5", classes=["center", "incident"])
            central_line(model, X_1, X_5, 1, 5, incident=True)
            X_35 = eval_trilinear_point(model, A, B, C, '1 + 2*cos(A)', ID="X_35", classes=["center", "incident"])
            X_546 = eval_trilinear_point(model, A, B, C, '3*cos(B - C) - 2*cos(A)', ID="X_546", classes=["center", "incident"])
            central_line(model, X_35, X_546, 35, 546, incident=True)
            X_381 = eval_trilinear_point(model, A, B, C, '2*cos(B - C) - cos(A)', ID="X_381", classes=["center", "incident"])
            X_498 = eval_trilinear_point(model, A, B, C, '1 + 2*sin(B)*sin(C)', ID="X_498", classes=["center", "incident"])
            central_line(model, X_381, X_498, 381, 498, incident=True)
            return promote_point_to_top(model, X_3614)


        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_3614 on the given reference triangle."""
            model = create_model('X_3614 = 1st HUTSON-FEUERBACH POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03614(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-03614.json"))
            write_center_svg(
                model_acute,
                here / "x-03614.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3614¶

X₃₆₁₄¶