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X₂₀₂₃¶

RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES

Let A'B'C' be the 1st Brocard triangle. Let A″ be the orthogonal projection of A on line B'C', and define B″ and C″ cyclically. Triangle A″B″C″ is perpsective to the medial triangle at X₂₀₂₃. (Note that the A-vertex of the medial triangle is the orthogonal projection of A' on line BC, and cyclically for B' and C'.) (Randy Hutson, July 11, 2019)

X₂₀₂₃ (RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES."""

        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,
            centroid_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            midpoint,
            nine_point_center_of,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_02023(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES on reference triangle ABC."""
            X_39 = eval_trilinear_point(model, A, B, C, 'a*(b**(2) + c**(2))', ID="X_39", classes=["center"])
            X_115 = eval_trilinear_point(model, A, B, C, 'b*c*(b**(2) - c**(2))**(2)', ID="X_115", classes=["center"])
            X_2023 = midpoint(model, X_39, X_115, ID="X_2023", classes=["selected", "target"], guide=False)
            X_5 = nine_point_center_of(model, A, B, C, ID="X_5", classes=["center", "incident"])
            central_line(model, X_5, X_39, 5, 39, incident=True)
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_98 = eval_trilinear_point(model, A, B, C, 'sec(A +  omega  )', ID="X_98", classes=["center", "incident"])
            central_line(model, X_6, X_98, 6, 98, incident=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_694 = eval_trilinear_point(model, A, B, C, 'a/(a**(4) - b**(2)*c**(2))', ID="X_694", classes=["center", "incident"])
            central_line(model, X_2, X_694, 2, 694, incident=True)
            return promote_point_to_top(model, X_2023)


        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_2023 on the given reference triangle."""
            model = create_model('X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_02023(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-02023.json"))
            write_center_svg(
                model_acute,
                here / "x-02023.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X2023¶

X₂₀₂₃¶