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X₁₈₄¶

INVERSE OF X₁₂₅ IN THE BROCARD CIRCLE

X₁₈₄ is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B₁ and C₁ be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C₂, A₂; A₃, B₃. Then A'B'C' is formed by the perpendicular bisectors of segments B₁C₁, C₂A₂, A₃B₃. (Fred Lang, Hyacinthos #1190)

X₁₈₄ (INVERSE OF X₁₂₅ IN THE BROCARD CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_184 = INVERSE OF X_125 IN THE BROCARD CIRCLE."""

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


        def construct_x_00184(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_184 = INVERSE OF X_125 IN THE BROCARD CIRCLE on reference triangle ABC."""
            X_184 = eval_trilinear_point(model, A, B, C, 'a**(2)*cos(A)', ID="X_184", classes=["selected", "target"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_25 = eval_trilinear_point(model, A, B, C, 'sin(A)*tan(A)', ID="X_25", classes=["center", "incident"])
            central_line(model, X_6, X_25, 6, 25, incident=True)
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_49 = eval_trilinear_point(model, A, B, C, 'cos(3*A)', ID="X_49", classes=["center", "incident"])
            central_line(model, X_3, X_49, 3, 49, incident=True)
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            X_54 = eval_trilinear_point(model, A, B, C, 'sec(B - C)', ID="X_54", classes=["center", "incident"])
            central_line(model, X_4, X_54, 4, 54, incident=True)
            return promote_point_to_top(model, X_184)


        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_184 on the given reference triangle."""
            model = create_model('X_184 = INVERSE OF X_125 IN THE BROCARD CIRCLE')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00184(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-00184.json"))
            write_center_svg(
                model_acute,
                here / "x-00184.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X184¶

X₁₈₄¶