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X₃₅₉₈¶

1st LIU POINT

Let O_A be the circle passing through B and C, internally tangent to the incircle, and likewise for O_B and O_C. Let P_A be the point where O_A meets the incircle, and likewise for P_B and P_C. Let Q_A be the point of intersection of the tangents to the incircle at P_B and P_C, and likewise for P_B and P_C. Then X₃₅₉₈ is the perspector of the triangles P_AP_BP_C and Q_AQ_BQ_C; this point is also X₇-of-Q_AQ_BQ_C. This point and X₃₅₉₉ were discovered by Kang-Ying Liu of St. Andrew’s Priory School, Honolulu, Hawaii, during 2010.

X₃₅₉₈ (1st LIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3598 = 1st 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_03598(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3598 = 1st LIU POINT on reference triangle ABC."""
            X_3598 = eval_trilinear_point(model, A, B, C, 'b*c*(a - b + c)*(a + b - c)*(3*a**(2) + b**(2) + c**(2) - 2*b*c)', ID="X_3598", 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_56 = eval_trilinear_point(model, A, B, C, 'a/(b + c - a)', ID="X_56", classes=["center", "incident"])
            X_105 = eval_trilinear_point(model, A, B, C, '1/(b**(2) + c**(2) - a*(b + c))', ID="X_105", classes=["center", "incident"])
            central_line(model, X_56, X_105, 56, 105, incident=True)
            X_22 = eval_trilinear_point(model, A, B, C, 'a*(b**(4) + c**(4) - a**(4))', ID="X_22", classes=["center", "incident"])
            X_347 = eval_trilinear_point(model, A, B, C, '((-1 - cos(A) + cos(B) + cos(C))*(sec(A/2))**(2))', ID="X_347", classes=["center", "incident"])
            central_line(model, X_22, X_347, 22, 347, incident=True)
            return promote_point_to_top(model, X_3598)


        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_3598 on the given reference triangle."""
            model = create_model('X_3598 = 1st LIU POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03598(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-03598.json"))
            write_center_svg(
                model_acute,
                here / "x-03598.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3598¶

X₃₅₉₈¶