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X₅₀₁¶

MIQUEL ASSOCIATE OF INCENTER

Miquel’s theorem states that if A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circles AB'C', BC'A', CA'B' meet at a point. Suppose P is a point and A' = P∩BC, B' = P∩CA, C' = P∩AB; the point in which the three circles is the Miquel associate of P. (Paul Yiu, 7/6/99)

X₅₀₁ (MIQUEL ASSOCIATE OF INCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_501 = MIQUEL ASSOCIATE OF INCENTER."""

        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,
            incenter_of,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00501(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_501 = MIQUEL ASSOCIATE OF INCENTER on reference triangle ABC."""
            X_501 = eval_trilinear_point(model, A, B, C, '(a*(a**(3) - b**(3) - c**(3) - b*c*(a + b + c) + a*b*(a - b) + a*c*(a - c))/(b + c))', ID="X_501", classes=["selected", "target"])
            X_36 = eval_trilinear_point(model, A, B, C, '1 - 2*cos(A)', ID="X_36", classes=["center", "incident"])
            X_58 = eval_trilinear_point(model, A, B, C, 'a/(b + c)', ID="X_58", classes=["center", "incident"])
            central_line(model, X_36, X_58, 36, 58, incident=True)
            X_35 = eval_trilinear_point(model, A, B, C, '1 + 2*cos(A)', ID="X_35", classes=["center", "incident"])
            X_110 = eval_trilinear_point(model, A, B, C, 'csc(B - C)', ID="X_110", classes=["center", "incident"])
            central_line(model, X_35, X_110, 35, 110, incident=True)
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_229 = point_from_barycentrics(model, A, B, C, 0.21061481, 0.29086323, 0.49852195, ID="X_229", classes=["center", "incident"])
            central_line(model, X_1, X_229, 1, 229, incident=True)
            return promote_point_to_top(model, X_501)


        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_501 on the given reference triangle."""
            model = create_model('X_501 = MIQUEL ASSOCIATE OF INCENTER')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00501(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-00501.json"))
            write_center_svg(
                model_acute,
                here / "x-00501.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X501¶

X₅₀₁¶