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X₁₇₃¶

CONGRUENT ISOSCELIZERS POINT

Let P_B on sideline AC and Q_C be equidistant from A, so that AP_BQ_C is an isosceles triangle. The line P_BQ_C is called an isoscelizer. The lines P_BQ_C, P_CQ_A, P_AQ_B concur in X₁₇₃. (P. Yff, unpublished notes, 1989)

X₁₇₃ (CONGRUENT ISOSCELIZERS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_173 = CONGRUENT ISOSCELIZERS 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,
            mittenpunkt_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00173(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_173 = CONGRUENT ISOSCELIZERS POINT on reference triangle ABC."""
            X_173 = eval_trilinear_point(model, A, B, C, 'cos(B/2) + cos(C/2) - cos(A/2)', ID="X_173", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_168 = eval_trilinear_point(model, A, B, C, 'sin(A) - sin(B) - sin(C) + 2*(cos(A/2) + sin(B/2 - A/2) + sin(C/2 - A/2))', ID="X_168", classes=["center", "incident"])
            central_line(model, X_1, X_168, 1, 168, incident=True)
            X_9 = mittenpunkt_of(model, A, B, C, ID="X_9", classes=["center", "incident"])
            X_177 = eval_trilinear_point(model, A, B, C, '(cos(B/2) + cos(C/2))*sec(A/2)', ID="X_177", classes=["center", "incident"])
            central_line(model, X_9, X_177, 9, 177, incident=True)
            X_57 = eval_trilinear_point(model, A, B, C, '1/(b + c - a)', ID="X_57", classes=["center", "incident"])
            X_174 = eval_trilinear_point(model, A, B, C, 'sec(A/2)', ID="X_174", classes=["center", "incident"])
            central_line(model, X_57, X_174, 57, 174, incident=True)
            return promote_point_to_top(model, X_173)


        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_173 on the given reference triangle."""
            model = create_model('X_173 = CONGRUENT ISOSCELIZERS POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00173(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-00173.json"))
            write_center_svg(
                model_acute,
                here / "x-00173.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X173¶

X₁₇₃¶