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X₂₉₀₄¶

ORTHIC-ISOGONAL CONJUGATE OF X₂₄

Let P be an arbitrary point of the circumcircle , O, and let S(P) be the Simson-Wallace line of P, and P' the reflection of P in S(P). The locus of P' as P ranges through O is a bicircular circumquartic that meets the sidelines of ABC in 6 points other than A, B, C. The 6 points lie on a conic, here called the Montesdeoca conic, M. The center of M is X₂₉₀₄, and the perspector of M is X₁₄₁₁₁. Let u = \(S_{A}\), v = \(S_{B}\), w = \(S_{C}\), \(f(a,b,c) = u^{2}(S^{2} - v^{2})(S^{2} - w^{2})\), \(g(a,b,c) = 2vw(u^{2} - S^{2})(S^{2} + vw)\). A barycentric equation for M is

X₂₉₀₄ (ORTHIC-ISOGONAL CONJUGATE OF X₂₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_2904 = ORTHIC-ISOGONAL CONJUGATE OF X_24."""

        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,
            ceva_conjugate,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            orthocenter_of,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_02904(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_2904 = ORTHIC-ISOGONAL CONJUGATE OF X_24 on reference triangle ABC."""
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center"])
            X_24 = eval_trilinear_point(model, A, B, C, 'sec(A)*cos(2*A)', ID="X_24", classes=["center"])
            X_2904 = ceva_conjugate(model, A, B, C, X_4, X_24, ID="X_2904", classes=["selected", "target"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_70 = eval_trilinear_point(model, A, B, C, 'b*c/(b**(2)*cos(2*B) + c**(2)*cos(2*C) - a**(2)*cos(2*A))', ID="X_70", classes=["center", "incident"])
            central_line(model, X_6, X_70, 6, 70, incident=True)
            X_52 = eval_trilinear_point(model, A, B, C, 'cos(2*A)*cos(B - C)', ID="X_52", classes=["center", "incident"])
            central_line(model, X_24, X_52, 24, 52, incident=True)
            X_25 = eval_trilinear_point(model, A, B, C, 'sin(A)*tan(A)', ID="X_25", classes=["center", "incident"])
            X_195 = eval_trilinear_point(model, A, B, C, 'a*(a**(8) + b**(8) + c**(8) - 4*a**(6)*(b**(2) + c**(2)) + a**(4)*(6*b**(4) + 6*c**(4) + 5*b**(2)*c**(2)) - a**(2)*(4*b**(6) + 4*c**(6) - b**(4)*c**(2) - b**(2)*c**(4)) - 2*b**(2)*c**(2)*(b**(4) + c**(4) - b**(2)*c**(2)))', ID="X_195", classes=["center", "incident"])
            central_line(model, X_25, X_195, 25, 195, incident=True)
            return promote_point_to_top(model, X_2904)


        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_2904 on the given reference triangle."""
            model = create_model('X_2904 = ORTHIC-ISOGONAL CONJUGATE OF X_24')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_02904(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-02904.json"))
            write_center_svg(
                model_acute,
                here / "x-02904.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X2904¶

X₂₉₀₄¶