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X₄₀₂¶

ZEEMAN-GOSSARD PERSPECTOR

In A History of Mathematics, Florian Cajori writes, “H. C. Gossard of the University of Oklahoma showed in 1916 that the three Euler lines of the triangles formed by the Euler line and the sides, taken by twos, of a given triangle, form a triangle … perspective with the given triangle and having the same Euler line.” Let ABC be the given triangle and A'B'C' the Gossard triangle - that is, the triangle perspective with the given triangle and having the same Euler line. The lines AA', BB', CC' concur in X₄₀₂, named the Gosssard perspector by John Conway (1998).

X₄₀₂ (ZEEMAN-GOSSARD PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_402 = ZEEMAN-GOSSARD PERSPECTOR."""

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


        def construct_x_00402(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_402 = ZEEMAN-GOSSARD PERSPECTOR on reference triangle ABC."""
            X_402 = eval_barycentric_point(model, A, B, C, '(2*a**(4) - a**(2)*b**(2) - b**(4) - a**(2)*c**(2) + 2*b**(2)*c**(2) - c**(4)) * (a**(8) - a**(6)*b**(2) - 2*a**(4)*b**(4) + 3*a**(2)*b**(6) - b**(8) - a**(6)*c**(2) + 5*a**(4)*b**(2)*c**(2) - 3*a**(2)*b**(4)*c**(2) - b**(6)*c**(2) - 2*a**(4)*c**(4) - 3*a**(2)*b**(2)*c**(4) + 4*b**(4)*c**(4) + 3*a**(2)*c**(6) - b**(2)*c**(6) - c**(8))', ID="X_402", 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_402)


        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_402 on the given reference triangle."""
            model = create_model('X_402 = ZEEMAN-GOSSARD PERSPECTOR')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00402(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-00402.json"))
            write_center_svg(
                model_acute,
                here / "x-00402.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X402¶

X₄₀₂¶