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X₁₂₈₅¶

LEMOINE HOMOTHETIC CENTER

X₁₂₈₅ is the homothetic center of the antipedal triangle of X₂ and the pedal triangle of X₆. (Darij Grinberg, Hyacinthos #6577, 2/21/03). See also X₃₀₆₆.

X₁₂₈₅ (LEMOINE HOMOTHETIC CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1285 = LEMOINE HOMOTHETIC CENTER."""

        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_barycentric_point,
            eval_trilinear_point,
            orthocenter_of,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01285(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1285 = LEMOINE HOMOTHETIC CENTER on reference triangle ABC."""
            X_1285 = eval_trilinear_point(model, A, B, C, '(b*c*(3*a**(2) + b**(2) - c**(2))*(3*a**(2) - b**(2) + c**(2)))', ID="X_1285", classes=["selected", "target"])
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            X_32 = eval_trilinear_point(model, A, B, C, 'a**(3)', ID="X_32", classes=["center", "incident"])
            central_line(model, X_4, X_32, 4, 32, incident=True)
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_376 = eval_trilinear_point(model, A, B, C, '5*cos(A) - cos(B - C)', ID="X_376", classes=["center", "incident"])
            central_line(model, X_6, X_376, 6, 376, incident=True)
            X_497 = eval_trilinear_point(model, A, B, C, '1 - cos(B)*cos(C)', ID="X_497", classes=["center", "incident"])
            X_609 = eval_barycentric_point(model, A, B, C, 'a*( area + a**(2)*sin(A))', ID="X_609", classes=["center", "incident"])
            central_line(model, X_497, X_609, 497, 609, incident=True)
            return promote_point_to_top(model, X_1285)


        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_1285 on the given reference triangle."""
            model = create_model('X_1285 = LEMOINE HOMOTHETIC CENTER')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01285(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-01285.json"))
            write_center_svg(
                model_acute,
                here / "x-01285.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1285¶

X₁₂₈₅¶