← etc / Encyclopedia of Triangle Centers

X₁₁₂₂¶

1st GRINBERG POINT

Let A_B be the touchpoint of the A-excircle and line AB, let A_C be the touchpoint of the A-excircle and line AC, and let M_A be the midpoint of segment A_BA_C. Define M_B and M_C cyclically. Let A', B', C' be the touchpoints of the incircle with lines BC, CA, AB, respectively. The triangles M_AM_BM_C and A'B'C' are perspective, and the perspector is X₁₁₂₂. (Darij Grinberg, 12/28/02)

X₁₁₂₂ (1st GRINBERG POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1122 = 1st GRINBERG 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,
            gergonne_point_of,
            nagel_point_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01122(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1122 = 1st GRINBERG POINT on reference triangle ABC."""
            X_1122 = eval_trilinear_point(model, A, B, C, '((a*(b + c) + (b - c)**(2))/(b + c - a))', ID="X_1122", classes=["selected", "target"])
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            X_8 = nagel_point_of(model, A, B, C, ID="X_8", classes=["center", "incident"])
            central_line(model, X_7, X_8, 7, 8, incident=True)
            X_56 = eval_trilinear_point(model, A, B, C, 'a/(b + c - a)', ID="X_56", classes=["center", "incident"])
            X_269 = eval_trilinear_point(model, A, B, C, '(tan(A/2))**(2)', ID="X_269", classes=["center", "incident"])
            central_line(model, X_56, X_269, 56, 269, incident=True)
            return promote_point_to_top(model, X_1122)


        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_1122 on the given reference triangle."""
            model = create_model('X_1122 = 1st GRINBERG POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01122(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-01122.json"))
            write_center_svg(
                model_acute,
                here / "x-01122.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1122¶

X₁₁₂₂¶