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X₃₅₇₂¶

INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The pair of central lines P(6)U(6) and P(8)U(8) concur in X₃₅₇₂. Contributed by Peter Moses, July 7, 2009.

X₃₅₇₂ (INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3572 = INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)."""

        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,
            isogonal_conjugate,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03572(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3572 = INTERSECTION OF LINES P(6)U(6) AND P(8)U(8) on reference triangle ABC."""
            X_3570 = eval_trilinear_point(model, A, B, C, 'b*c*(a**(2) - b*c)/(b - c)', ID="X_3570", classes=["center"])
            X_3572 = isogonal_conjugate(model, A, B, C, X_3570, ID="X_3572", classes=["selected", "target"])
            X_37 = eval_trilinear_point(model, A, B, C, 'b + c', ID="X_37", classes=["center", "incident"])
            X_513 = eval_trilinear_point(model, A, B, C, 'b - c', ID="X_513", classes=["center", "incident"])
            central_line(model, X_37, X_513, 37, 513, incident=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_661 = eval_trilinear_point(model, A, B, C, 'cot(B) - cot(C)', ID="X_661", classes=["center", "incident"])
            central_line(model, X_2, X_661, 2, 661, incident=True)
            X_42 = eval_trilinear_point(model, A, B, C, 'a*(b + c)', ID="X_42", classes=["center", "incident"])
            X_649 = eval_trilinear_point(model, A, B, C, 'a*(b - c)', ID="X_649", classes=["center", "incident"])
            central_line(model, X_42, X_649, 42, 649, incident=True)
            return promote_point_to_top(model, X_3572)


        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_3572 on the given reference triangle."""
            model = create_model('X_3572 = INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03572(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-03572.json"))
            write_center_svg(
                model_acute,
                here / "x-03572.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3572¶

X₃₅₇₂¶