← etc / Encyclopedia of Triangle Centers

X₁₁₀₈¶

CROSSPOINT OF X₂ AND X₈₄

X₁₁₀₈ is the midpoint of the bicentric pair \(y : z : x\) and \(z : x : y\), where \(x : y : z\) = X₂₁₉

X₁₁₀₈ (CROSSPOINT OF X₂ AND X₈₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1108 = CROSSPOINT OF X_2 AND X_84."""

        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,
            complement,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            incenter_of,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01108(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1108 = CROSSPOINT OF X_2 AND X_84 on reference triangle ABC."""
            X_322 = eval_trilinear_point(model, A, B, C, '((-1 - cos(A) + cos(B) + cos(C))*(csc(A))**(2))', ID="X_322", classes=["center"])
            X_1108 = complement(model, A, B, C, X_322, ID="X_1108", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            central_line(model, X_1, X_6, 1, 6, incident=True)
            X_19 = eval_trilinear_point(model, A, B, C, 'tan(A)', ID="X_19", classes=["center", "incident"])
            X_56 = eval_trilinear_point(model, A, B, C, 'a/(b + c - a)', ID="X_56", classes=["center", "incident"])
            central_line(model, X_19, X_56, 19, 56, incident=True)
            X_104 = eval_trilinear_point(model, A, B, C, '1/(-1 + cos(B) + cos(C))', ID="X_104", classes=["center", "incident"])
            X_112 = eval_trilinear_point(model, A, B, C, 'a/(sin(2*B) - sin(2*C))', ID="X_112", classes=["center", "incident"])
            central_line(model, X_104, X_112, 104, 112, incident=True)
            return promote_point_to_top(model, X_1108)


        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_1108 on the given reference triangle."""
            model = create_model('X_1108 = CROSSPOINT OF X_2 AND X_84')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01108(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-01108.json"))
            write_center_svg(
                model_acute,
                here / "x-01108.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1108¶

X₁₁₀₈¶