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X₁₉₉₅¶

ORTHOCORRESPONDENT OF X₆

Let P be a point in the plane of ABC and not on the circumcircle. Let (O_A) be the circle tangent to the circumcircle at A and passing through P. Let A' be the antipode of A in (O_A). Let L_A be the tangent to (O_A) at A'. Define L_B and L_C cyclically. Let T_A = L_B∩L_C, and define T_B and T_C cyclically. Triangle T_AT_BT_C is homothetic to the tangential triangle. When P = X₆, the center of homothety is X₁₉₉₅. (Randy Hutson, June 7, 2019)

X₁₉₉₅ (ORTHOCORRESPONDENT OF X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1995 = ORTHOCORRESPONDENT OF X_6."""

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


        def construct_x_01995(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1995 = ORTHOCORRESPONDENT OF X_6 on reference triangle ABC."""
            X_598 = eval_trilinear_point(model, A, B, C, 'b*c/(a**(2) - 2*b**(2) - 2*c**(2))', ID="X_598", classes=["center"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center"])
            X_1995 = ceva_conjugate(model, A, B, C, X_598, X_6, ID="X_1995", 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)
            X_110 = eval_trilinear_point(model, A, B, C, 'csc(B - C)', ID="X_110", classes=["center", "incident"])
            central_line(model, X_6, X_110, 6, 110, incident=True)
            X_107 = eval_trilinear_point(model, A, B, C, '1/(cos(A)*(sin(2*B) - sin(2*C)))', ID="X_107", classes=["center", "incident"])
            X_264 = eval_trilinear_point(model, A, B, C, 'csc(A)*csc(2*A)', ID="X_264", classes=["center", "incident"])
            central_line(model, X_107, X_264, 107, 264, incident=True)
            return promote_point_to_top(model, X_1995)


        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_1995 on the given reference triangle."""
            model = create_model('X_1995 = ORTHOCORRESPONDENT OF X_6')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01995(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-01995.json"))
            write_center_svg(
                model_acute,
                here / "x-01995.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1995¶

X₁₉₉₅¶