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X₁₁₄₄¶

EHRMANN CONGRUENT SQUARES POINT

Suppose that P is a point inside triangle ABC. Let \(S_{A}\) be the square inscribed in triangle PBC, having two vertices on segment BC, one on PB, and one on PC. Define \(S_{B}\) and \(S_{C}\) cyclically. Then X₁₁₄₄ is the unique choice of P for which the three squares are congruent. The function L(a,b,c) is symmetric, homogeneous of degree 1, and satisfies 0 < L(a,b,c) < min{a,b,c}. Also, X₁₁₄₄ lies on the hyperbola {A, B, C,X₁,X₆}; indeed, X₁₁₄₄ lies on the open arc from X₁ to the vertex of ABC opposite the shortest side. L(a,b,c) is the common length of the sides of the three squares. (Jean-Pierre Ehrmann, 12/16/01)

X₁₁₄₄ (EHRMANN CONGRUENT SQUARES POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1144 = EHRMANN CONGRUENT SQUARES POINT."""

        from __future__ import annotations

        from pathlib import Path

        import sympy.geometry as spg
        from geometor.model import Model

        from geometor.etc.ops import (
            construct_baseline_triangle,
            create_model,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01144(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1144 = EHRMANN CONGRUENT SQUARES POINT on reference triangle ABC."""
            X_1144 = point_from_barycentrics(model, A, B, C, 0.29332424, 0.33285663, 0.37381913, ID="X_1144", classes=["selected", "target"])
            return promote_point_to_top(model, X_1144)


        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_1144 on the given reference triangle."""
            model = create_model('X_1144 = EHRMANN CONGRUENT SQUARES POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01144(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-01144.json"))
            write_center_svg(
                model_acute,
                here / "x-01144.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1144¶

X₁₁₄₄¶