← etc / Encyclopedia of Triangle Centers

X₃¶

CIRCUMCENTER

X₃ is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X₃. This common distance is the radius of the circumcircle, which passes through vertices A, B, C. Called the circumradius, it is given by \(R = a/(2 \sin A) = abc/(4 \operatorname{area}(ABC))\).

X₃ (CIRCUMCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3 = 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,
            centroid_of,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            orthocenter_of,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00003(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3 = CIRCUMCENTER on reference triangle ABC."""
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["selected", "target"], draw_bisectors=True, draw_circumcircle=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center"])
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center"])
            central_line(model, X_2, X_4, 2, 4)
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_15 = eval_trilinear_point(model, A, B, C, 'sin(A +  pi  /3)', ID="X_15", classes=["center", "incident"])
            central_line(model, X_6, X_15, 6, 15, incident=True)
            X_13 = eval_trilinear_point(model, A, B, C, 'csc(A +  pi  /3)', ID="X_13", classes=["center", "incident"])
            X_17 = eval_trilinear_point(model, A, B, C, 'csc(A +  pi  /6)', ID="X_17", classes=["center", "incident"])
            central_line(model, X_13, X_17, 13, 17, incident=True)
            return promote_point_to_top(model, X_3)


        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_3 on the given reference triangle."""
            model = create_model('X_3 = CIRCUMCENTER')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00003(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-00003.json"))
            write_center_svg(
                model_acute,
                here / "x-00003.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3¶

X₃¶