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X₉₀₃¶

X₁-BLAIKIE TRANSFORM OF X₂

James Blaikie (1847-1929) proposed the following problem. Let O be any point in the plane of triangle ABC, and let any straight line g through O meet BC in P, CA in Q, AB in R; then, if points P', Q', R' be taken on the line so that PO = OP', QO = OQ', RO = OR'. Prove that AP', BQ', CR' concur.

X₉₀₃ (X₁-BLAIKIE TRANSFORM OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_903 = X_1-BLAIKIE TRANSFORM OF X_2."""

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


        def construct_x_00903(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_903 = X_1-BLAIKIE TRANSFORM OF X_2 on reference triangle ABC."""
            X_902 = eval_trilinear_point(model, A, B, C, 'a*(b + c - 2*a)', ID="X_902", classes=["center"])
            X_903 = isogonal_conjugate(model, A, B, C, X_902, ID="X_903", classes=["selected", "target"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_45 = eval_trilinear_point(model, A, B, C, '2*b + 2*c - a', ID="X_45", classes=["center", "incident"])
            central_line(model, X_2, X_45, 2, 45, incident=True)
            X_86 = eval_trilinear_point(model, A, B, C, 'b*c/(b + c)', ID="X_86", classes=["center", "incident"])
            X_99 = eval_trilinear_point(model, A, B, C, 'b*c/(b**(2) - c**(2))', ID="X_99", classes=["center", "incident"])
            central_line(model, X_86, X_99, 86, 99, incident=True)
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            X_528 = point_from_barycentrics(model, A, B, C, -0.26726124, 0.80178373, -0.53452248, ID="X_528", classes=["center", "incident"])
            central_line(model, X_7, X_528, 7, 528, incident=True)
            return promote_point_to_top(model, X_903)


        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_903 on the given reference triangle."""
            model = create_model('X_903 = X_1-BLAIKIE TRANSFORM OF X_2')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00903(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-00903.json"))
            write_center_svg(
                model_acute,
                here / "x-00903.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X903¶

X₉₀₃¶