← etc / Encyclopedia of Triangle Centers

X₇₃¶

CROSSPOINT OF INCENTER AND CIRCUMCENTER

X₇₃ is the crosspoint of incenter and circumcenter.

X₇₃ (CROSSPOINT OF INCENTER AND CIRCUMCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER."""

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


        def construct_x_00073(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER on reference triangle ABC."""
            X_29 = eval_trilinear_point(model, A, B, C, '(sec(A))/(cos(B) + cos(C))', ID="X_29", classes=["center"])
            X_73 = isogonal_conjugate(model, A, B, C, X_29, ID="X_73", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center", "incident"])
            central_line(model, X_1, X_4, 1, 4, incident=True)
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_41 = eval_trilinear_point(model, A, B, C, 'a**(2)*(b + c - a)', ID="X_41", classes=["center", "incident"])
            central_line(model, X_6, X_41, 6, 41, incident=True)
            X_36 = eval_trilinear_point(model, A, B, C, '1 - 2*cos(A)', ID="X_36", 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_36, X_54, 36, 54, incident=True)
            return promote_point_to_top(model, X_73)


        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_73 on the given reference triangle."""
            model = create_model('X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00073(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-00073.json"))
            write_center_svg(
                model_acute,
                here / "x-00073.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X73¶

X₇₃¶