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X₁₃₃₉¶

NAGEL-SCHRÖDER POINT

Let N_aN_bN_c be the extouch triangle of ABC; viz., N_a is where the A-excircle meets line BC, and N_aN_bN_c is the pedal triangle of X₄₀. Let I = incenter of ABC. The circles (AIX’), (BIY’), (CIZ’) concur in two points: I and X₁₃₃₉. (Jean-Pierre Ehrmann, Hyacinthos #6545)

X₁₃₃₉ (NAGEL-SCHRÖDER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1339 = NAGEL-SCHRÖDER 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,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01339(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1339 = NAGEL-SCHRÖDER POINT on reference triangle ABC."""
            X_1339 = eval_trilinear_point(model, A, B, C, '((b+c-2*a)*(4*b*c*(b+c-a) - (a+b+c)*(b**(2) + c**(2) - a**(2)))/(b+c-3*a))', ID="X_1339", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_474 = eval_trilinear_point(model, A, B, C, 'cos(A) - (a + b + c)/a', ID="X_474", classes=["center", "incident"])
            central_line(model, X_1, X_474, 1, 474, incident=True)
            return promote_point_to_top(model, X_1339)


        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_1339 on the given reference triangle."""
            model = create_model('X_1339 = NAGEL-SCHRÖDER POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01339(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-01339.json"))
            write_center_svg(
                model_acute,
                here / "x-01339.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1339¶

X₁₃₃₉¶