← etc / Encyclopedia of Triangle Centers

X₄₉¶

CENTER OF SINE-TRIPLE-ANGLE CIRCLE

Suppose that P and Q are distinct points in the plane of a triangle ABC . Let P_A = reflection of P in line AQ, let Q_A = reflection of Q in line AP, and let M_A = midpoint of segment P_AQ_A. Define M_B and M_C cyclically. César Lozada found that if Q = isogonal conjugate of P, then the locus of P for which M_AM_BM_C is perspective to ABC is the union of a cubic and 6 circles: specifically, the McCay cubic (K003), the circles {B,C,J_b,J_c}}, {C,A,J_c,J_a}}, {A,B,J_a,J_b}}, and the circles {B,C,J_a}}, {C,A,J_b}}, {A,B,J_c}}, where J_a,J_b,J_c are the excenters of ABC. Moreover, if P = X₃ and Q = X₄, then M_AM_BM_C is not only perspective, but homothetic, to ABC, and the center of homothety is X₄₉. See Hyacinthos 23265, June 1, 2015.

X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE."""

        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,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            eval_barycentric_point,
            eval_trilinear_point,
            nine_point_center_of,
            orthocenter_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00049(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE on reference triangle ABC."""
            X_49 = eval_trilinear_point(model, A, B, C, 'cos(3*A)', ID="X_49", classes=["selected", "target"])
            X_5 = nine_point_center_of(model, A, B, C, ID="X_5", classes=["center", "incident"])
            X_54 = eval_trilinear_point(model, A, B, C, 'sec(B - C)', ID="X_54", classes=["center", "incident"])
            central_line(model, X_5, X_54, 5, 54, incident=True)
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_155 = eval_trilinear_point(model, A, B, C, '(cos(A))*((cos(B))**(2) + (cos(C))**(2) - (cos(A))**(2))', ID="X_155", classes=["center", "incident"])
            central_line(model, X_3, X_155, 3, 155, incident=True)
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            X_156 = eval_barycentric_point(model, A, B, C, 'a**(2)*(a**(8) - 3*a**(6)*(b**(2) + c**(2)) + a**(4)*(3*b**(4) + 2*b**(2)*c**(2) + 3*c**(4)) - a**(2)*(b**(6) + c**(6)) + b**(2)*c**(2)*(b**(2) - c**(2))**(2))', ID="X_156", classes=["center", "incident"])
            central_line(model, X_4, X_156, 4, 156, incident=True)
            return promote_point_to_top(model, X_49)


        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_49 on the given reference triangle."""
            model = create_model('X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00049(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-00049.json"))
            write_center_svg(
                model_acute,
                here / "x-00049.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X49¶

X₄₉¶