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X₃₀₈₂¶

RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES

Let (O_A) be the I-excircle of triangle IBC, where I = X₁, and define (O_B) and (O_C) cyclically. Let A' be the touchpoint of (O_A) and side BC, and define B' and C' cyclically. Let A″ be the insimilicenter of (O_B) and (O_C), and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₃₀₈₂. (Randy Hutson, July 11, 2019)

X₃₀₈₂ (RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3082 = RADICAL CENTER OF EXTERNAL MALFATTI 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,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            nagel_point_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03082(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3082 = RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES on reference triangle ABC."""
            X_3082 = eval_trilinear_point(model, A, B, C, '(1/(1 - cos(A/2)))', ID="X_3082", classes=["selected", "target"])
            X_8 = nagel_point_of(model, A, B, C, ID="X_8", classes=["center", "incident"])
            X_178 = eval_trilinear_point(model, A, B, C, '(cos(B/2) + cos(C/2))*csc(A)', ID="X_178", classes=["center", "incident"])
            central_line(model, X_8, X_178, 8, 178, incident=True)
            X_174 = eval_trilinear_point(model, A, B, C, 'sec(A/2)', ID="X_174", classes=["center", "incident"])
            X_176 = eval_trilinear_point(model, A, B, C, '1 + sec(A/2)*cos(B/2)*cos(C/2)', ID="X_176", classes=["center", "incident"])
            central_line(model, X_174, X_176, 174, 176, incident=True)
            return promote_point_to_top(model, X_3082)


        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_3082 on the given reference triangle."""
            model = create_model('X_3082 = RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03082(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-03082.json"))
            write_center_svg(
                model_acute,
                here / "x-03082.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3082¶

X₃₀₈₂¶