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X₁₃₃₈¶

2nd WERNAU POINT

Let A'BC be the internal equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X₁₃₃₈. For details, see X₁₃₃₇.

X₁₃₃₈ (2nd WERNAU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1338 = 2nd WERNAU 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,
            construct_baseline_triangle,
            create_model,
            eval_barycentric_point,
            eval_trilinear_point,
            orthocenter_of,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01338(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1338 = 2nd WERNAU POINT on reference triangle ABC."""
            X_1338 = eval_barycentric_point(model, A, B, C, '(a**(2)*(2*D*U + 3**(1/2)*V*W)/(4*D - 3**(1/2)*a**(2)))', ID="X_1338", classes=["selected", "target"])
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            X_617 = point_from_barycentrics(model, A, B, C, 0.67910401, -1.0602234, 1.3811194, ID="X_617", classes=["center", "incident"])
            central_line(model, X_4, X_617, 4, 617, incident=True)
            X_1157 = eval_trilinear_point(model, A, B, C, '4*cos(A) + cos(3*A)*sec(A)*sec(B - C)', ID="X_1157", classes=["center", "incident"])
            X_1337 = eval_barycentric_point(model, A, B, C, '(a**(2)*(2*D*U - 3**(1/2)*V*W)/(4*D + 3**(1/2)*a**(2)))', ID="X_1337", classes=["center", "incident"])
            central_line(model, X_1157, X_1337, 1157, 1337, incident=True)
            X_16 = eval_trilinear_point(model, A, B, C, 'sin(A-  pi  /3)', ID="X_16", classes=["center", "incident"])
            X_3458 = eval_trilinear_point(model, A, B, C, '(((sin(A))**(2))*(sin(A/2))*sec(A +  pi  /6))', ID="X_3458", classes=["center", "incident"])
            central_line(model, X_16, X_3458, 16, 3458, incident=True)
            return promote_point_to_top(model, X_1338)


        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_1338 on the given reference triangle."""
            model = create_model('X_1338 = 2nd WERNAU POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01338(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-01338.json"))
            write_center_svg(
                model_acute,
                here / "x-01338.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1338¶

X₁₃₃₈¶