← etc / Encyclopedia of Triangle Centers

X₁₁₅₉¶

GREENHILL POINT

X₁₁₅₉ = {X₂₀₉₉,X₄₈₆₀}-harmonic conjugate of X₁

X₁₁₅₉ (GREENHILL POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1159 = GREENHILL 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,
            circumcenter_of,
            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_01159(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1159 = GREENHILL POINT on reference triangle ABC."""
            X_1159 = eval_trilinear_point(model, A, B, C, '(cos(A) + 4*cos(B) + 4*cos(C))', ID="X_1159", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            central_line(model, X_1, X_3, 1, 3, incident=True)
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            X_952 = eval_trilinear_point(model, A, B, C, 'b*c*(2*a**(4) - 2*a**(3)*(b + c) - a**(2)*(b**(2) - 4*b*c + c**(2)) + (2*a - b - c)*(b - c)*(b**(2) - c**(2)))', ID="X_952", classes=["center", "incident"])
            central_line(model, X_7, X_952, 7, 952, incident=True)
            return promote_point_to_top(model, X_1159)


        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_1159 on the given reference triangle."""
            model = create_model('X_1159 = GREENHILL POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01159(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-01159.json"))
            write_center_svg(
                model_acute,
                here / "x-01159.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1159¶

X₁₁₅₉¶