← etc / Encyclopedia of Triangle Centers

X₁₅₉₄¶

RIGBY-LALESCU ORTHOPOLE

Traian Lalescu (Trajan Lalesco) proved in “A Class of Remarcable Triangles,” Gazeta Matematica 20 (1915) 213 [in Romanian], that if triangles DEF and D'E'F' are inscribed in a circle and directed arclengths satisfyarc DD' + arc EE' + arc FF' = 0 mod 2π, then the Simson lines of D, E, F with respect to D',E',F' and the Simson lines of D',E',F' with respect to D, E, F concur in the midpoint X of the segment of the orthocenters of DEF and D'E'F'. Daniel Vacaretu considered triangles DEF and D'E'F' associated with left and right isoscelizers and inscribed in the sine-triple-angle circle. He obtained the second set of trilinears shown above for the midpoint X. (See also the bicentric pair PU(61).)

X₁₅₉₄ (RIGBY-LALESCU ORTHOPOLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1594 = RIGBY-LALESCU ORTHOPOLE."""

        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_barycentric_point,
            eval_trilinear_point,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01594(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1594 = RIGBY-LALESCU ORTHOPOLE on reference triangle ABC."""
            X_933 = eval_trilinear_point(model, A, B, C, 'sec(A)*csc(2*B - 2*C)', ID="X_933", classes=["center"])
            X_523 = eval_trilinear_point(model, A, B, C, 'sin(B -C)', ID="X_523", classes=["center"])
            X_1594 = ceva_conjugate(model, A, B, C, X_933, X_523, ID="X_1594", 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_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_128 = eval_trilinear_point(model, A, B, C, '((sec(A))*(cos(2*B) + cos(2*C))*(1 + 2*cos(2*A))*(cos(2*A) + 2*cos(2*B)*cos(2*C)))', ID="X_128", classes=["center", "incident"])
            X_136 = eval_barycentric_point(model, A, B, C, '(b**(2) - c**(2))**(2)*(a**(4) + b**(4) + c**(4) - 2*a**(2)*b**(2) - 2*a**(2)*c**(2))/(a**(2) - b**(2) - c**(2))', ID="X_136", classes=["center", "incident"])
            central_line(model, X_128, X_136, 128, 136, incident=True)
            return promote_point_to_top(model, X_1594)


        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_1594 on the given reference triangle."""
            model = create_model('X_1594 = RIGBY-LALESCU ORTHOPOLE')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01594(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-01594.json"))
            write_center_svg(
                model_acute,
                here / "x-01594.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1594¶

X₁₅₉₄¶