← etc / Encyclopedia of Triangle Centers

X₁₁₁₃¶

1st EULER-LINE-CIRCUMCIRCLE INTERSECTION

X₁₁₁₃ is a point of intersection of the Euler line and the circumcircle. The other is X₁₁₁₄. Of the two, X₁₁₁₃ is the one closer to X₄.

X₁₁₁₃ (1st EULER-LINE-CIRCUMCIRCLE INTERSECTION) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1113 = 1st EULER-LINE-CIRCUMCIRCLE INTERSECTION."""

        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,
            incenter_of,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01113(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1113 = 1st EULER-LINE-CIRCUMCIRCLE INTERSECTION on reference triangle ABC."""
            X_1113 = eval_trilinear_point(model, A, B, C, '(R - d)*cos(A) - 2*R*cos(B)*cos(C)', ID="X_1113", 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_109 = eval_trilinear_point(model, A, B, C, 'a/(cos(B) - cos(C))', ID="X_109", classes=["center", "incident"])
            X_1822 = point_from_barycentrics(model, A, B, C, -0.19858607, 0.24538924, 0.95319683, ID="X_1822", classes=["center", "incident"])
            central_line(model, X_109, X_1822, 109, 1822, incident=True)
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_2100 = point_from_barycentrics(model, A, B, C, -0.39033676, -0.15032397, 1.5406607, ID="X_2100", classes=["center", "incident"])
            central_line(model, X_1, X_2100, 1, 2100, incident=True)
            return promote_point_to_top(model, X_1113)


        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_1113 on the given reference triangle."""
            model = create_model('X_1113 = 1st EULER-LINE-CIRCUMCIRCLE INTERSECTION')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01113(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-01113.json"))
            write_center_svg(
                model_acute,
                here / "x-01113.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1113¶

X₁₁₁₃¶