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X₁₁₄₃¶

2nd MALFATTI-RABINOWITZ POINT

Along each side of ABC there is a segment that is a common tangent to two of the three Malfatti circles of ABC. Let A', B', C' be the midpoints of these respective segments. Then triangle A'B'C' isperspective to ABC, and the perspector is X₁₁₄₃. (Stanley Rabinowitz, #4611, 12/29/01) For coordinates, see Paul Yiu, #4615, 12/30/01, and

X₁₁₄₃ (2nd MALFATTI-RABINOWITZ POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1143 = 2nd MALFATTI-RABINOWITZ 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,
            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_01143(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1143 = 2nd MALFATTI-RABINOWITZ POINT on reference triangle ABC."""
            X_1143 = eval_trilinear_point(model, A, B, C, 'csc(A)*tan(A/4)', ID="X_1143", classes=["selected", "target"])
            X_8 = nagel_point_of(model, A, B, C, ID="X_8", classes=["center", "incident"])
            X_177 = eval_trilinear_point(model, A, B, C, '(cos(B/2) + cos(C/2))*sec(A/2)', ID="X_177", classes=["center", "incident"])
            central_line(model, X_8, X_177, 8, 177, incident=True)
            X_174 = eval_trilinear_point(model, A, B, C, 'sec(A/2)', ID="X_174", classes=["center", "incident"])
            X_175 = eval_trilinear_point(model, A, B, C, '-1 + sec(A/2)*cos(B/2)*cos(C/2)', ID="X_175", classes=["center", "incident"])
            central_line(model, X_174, X_175, 174, 175, incident=True)
            X_558 = eval_trilinear_point(model, A, B, C, 'sec(A/2)*tan(A/4)', ID="X_558", classes=["center", "incident"])
            X_1488 = eval_trilinear_point(model, A, B, C, '1/(1 + sin(A/2))', ID="X_1488", classes=["center", "incident"])
            central_line(model, X_558, X_1488, 558, 1488, incident=True)
            return promote_point_to_top(model, X_1143)


        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_1143 on the given reference triangle."""
            model = create_model('X_1143 = 2nd MALFATTI-RABINOWITZ POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01143(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-01143.json"))
            write_center_svg(
                model_acute,
                here / "x-01143.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1143¶

X₁₁₄₃¶