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X₁₁₃₉¶

OUTER PENTAGON POINT

Let A' be the outermost vertex of the regular pentagon erected outward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X₁₁₃₉. (Steve Sigur and Antreas Hatzipolakis, #5246, 12/31/02; generalizations to n-gons, for odd n, by Milorad R. Stevanovic, #5253, 5256)

X₁₁₃₉ (OUTER PENTAGON POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1139 = OUTER PENTAGON 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,
            orthocenter_of,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01139(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1139 = OUTER PENTAGON POINT on reference triangle ABC."""
            X_1139 = eval_trilinear_point(model, A, B, C, '(csc(A))/(cot(A) + cot(2*pi/5))', ID="X_1139", classes=["selected", "target"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_1140 = eval_trilinear_point(model, A, B, C, '(csc(A))/(cot(A) - cot(2*pi/5))', ID="X_1140", classes=["center", "incident"])
            central_line(model, X_6, X_1140, 6, 1140, incident=True)
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_3369 = eval_trilinear_point(model, A, B, C, '(sin(A - 2*pi  /5))', ID="X_3369", classes=["center", "incident"])
            central_line(model, X_1, X_3369, 1, 3369, incident=True)
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            X_3368 = eval_trilinear_point(model, A, B, C, '(cos(A - 2*pi  /5))', ID="X_3368", classes=["center", "incident"])
            central_line(model, X_4, X_3368, 4, 3368, incident=True)
            return promote_point_to_top(model, X_1139)


        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_1139 on the given reference triangle."""
            model = create_model('X_1139 = OUTER PENTAGON POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01139(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-01139.json"))
            write_center_svg(
                model_acute,
                here / "x-01139.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1139¶

X₁₁₃₉¶