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X₄₈₁¶

1st EPPSTEIN POINT

Let S be the inner Soddy circle and \(S_{A}\), \(S_{B}\), \(S_{C}\) the Soddy circles tangent to S. Let I_a = S∩\(S_{A}\), E_a = \(S_{B}\)∩\(S_{C}\), and determine I_b, I_c, E_b, E_c cyclically. Then X₄₈₁ is the point of concurrence of lines I_A-to-E_A, I_B-to-E_B, I_C-to-E_C.

X₄₈₁ (1st EPPSTEIN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_481 = 1st EPPSTEIN 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,
            incenter_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00481(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_481 = 1st EPPSTEIN POINT on reference triangle ABC."""
            X_481 = eval_trilinear_point(model, A, B, C, '1 - 2*sec(A/2)*cos(B/2)*cos(C/2)', ID="X_481", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            central_line(model, X_1, X_7, 1, 7, incident=True)
            X_226 = eval_trilinear_point(model, A, B, C, '(csc(A))*(cos(B) + cos(C))', ID="X_226", classes=["center", "incident"])
            X_485 = eval_trilinear_point(model, A, B, C, 'sec(A -  pi  /4)', ID="X_485", classes=["center", "incident"])
            central_line(model, X_226, X_485, 226, 485, incident=True)
            X_174 = eval_trilinear_point(model, A, B, C, 'sec(A/2)', ID="X_174", classes=["center", "incident"])
            X_1127 = eval_trilinear_point(model, A, B, C, '(sin(A/4))/sin(3*A/4)', ID="X_1127", classes=["center", "incident"])
            central_line(model, X_174, X_1127, 174, 1127, incident=True)
            return promote_point_to_top(model, X_481)


        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_481 on the given reference triangle."""
            model = create_model('X_481 = 1st EPPSTEIN POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00481(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-00481.json"))
            write_center_svg(
                model_acute,
                here / "x-00481.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X481¶

X₄₈₁¶