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X₃₅₆₈¶

BELTRAMI-EULER POINT

Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X₃₅₆₈ is the point of intersection of the lines P(2)P(40) and U(2)U(40). The name “Beltrami-Euler Point” signifies the fact that X₃₅₆₈ lies on the Euler line. See also X₂₉₆₆ and X₃₅₆₉. Contributed by Chris van Tienhoven, June 7, 2009.

X₃₅₆₈ (BELTRAMI-EULER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3568 = BELTRAMI-EULER 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,
            centroid_of,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03568(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3568 = BELTRAMI-EULER POINT on reference triangle ABC."""
            X_3568 = point_from_barycentrics(model, A, B, C, 0.38367849, 0.36457958, 0.25174194, ID="X_3568", classes=["selected", "target"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center"])
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center"])
            central_line(model, X_2, X_3, 2, 3)
            return promote_point_to_top(model, X_3568)


        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_3568 on the given reference triangle."""
            model = create_model('X_3568 = BELTRAMI-EULER POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03568(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-03568.json"))
            write_center_svg(
                model_acute,
                here / "x-03568.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3568¶

X₃₅₆₈¶