← etc / Encyclopedia of Triangle Centers

X₃₅₉₉¶

2nd LIU POINT

Let P_AP_BP_C be as at X₃₅₉₈. Let T_A be the point of intersection of the lines BP_C and CP_B. X₃₅₉₉ is the perspector of triangles P_AP_BP_C and T_AT_BT_C.

X₃₅₉₉ (2nd LIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3599 = 2nd LIU 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,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            gergonne_point_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03599(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3599 = 2nd LIU POINT on reference triangle ABC."""
            X_3599 = eval_trilinear_point(model, A, B, C, 'b*c*(a - b + c)*(a + b - c)*(5*a**(4) + b**(4) + c**(4) - 8*a**(3)*b - 8*a**(3)*c - 4*b**(3)*c - 4*b*c**(3) + 6*b**(2)*c**(2) + 2*a**(2)*b**(2) + 2*a**(2)*c**(2) + 12*a**(2)*b*c)', ID="X_3599", classes=["selected", "target"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            central_line(model, X_2, X_7, 2, 7, incident=True)
            X_165 = eval_trilinear_point(model, A, B, C, 'tan(B/2) + tan(C/2) - tan(A/2)', ID="X_165", classes=["center", "incident"])
            X_279 = eval_trilinear_point(model, A, B, C, 'csc(A)*(tan(A/2))**(2)', ID="X_279", classes=["center", "incident"])
            central_line(model, X_165, X_279, 165, 279, incident=True)
            X_55 = eval_trilinear_point(model, A, B, C, 'a*(b + c - a)', ID="X_55", classes=["center", "incident"])
            X_479 = eval_trilinear_point(model, A, B, C, '(tan(A/2)*sec(A/2))**(2)', ID="X_479", classes=["center", "incident"])
            central_line(model, X_55, X_479, 55, 479, incident=True)
            return promote_point_to_top(model, X_3599)


        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_3599 on the given reference triangle."""
            model = create_model('X_3599 = 2nd LIU POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03599(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-03599.json"))
            write_center_svg(
                model_acute,
                here / "x-03599.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3599¶

X₃₅₉₉¶