← etc / Encyclopedia of Triangle Centers

X₁₀₀₁¶

MIDPOINT OF X₁ AND X₉

Let A* be the tangency point of O_A and the circumcircle, and define B* and C* cyclically. Let A** be the radical trace of O_B and O_C. The lines A*A**, B*B**, C*C** concur in X₁₀₀₁.

X₁₀₀₁ (MIDPOINT OF X₁ AND X₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1001 = MIDPOINT OF X_1 AND X_9."""

        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,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            feuerbach_point_of,
            gergonne_point_of,
            incenter_of,
            midpoint,
            mittenpunkt_of,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01001(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1001 = MIDPOINT OF X_1 AND X_9 on reference triangle ABC."""
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center"])
            X_9 = mittenpunkt_of(model, A, B, C, ID="X_9", classes=["center"])
            X_1001 = midpoint(model, X_1, X_9, ID="X_1001", classes=["selected", "target"], guide=False)
            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_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_11 = feuerbach_point_of(model, A, B, C, ID="X_11", classes=["center", "incident"])
            central_line(model, X_2, X_11, 2, 11, incident=True)
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            X_21 = eval_trilinear_point(model, A, B, C, '1/(cos(B) + cos(C))', ID="X_21", classes=["center", "incident"])
            central_line(model, X_7, X_21, 7, 21, incident=True)
            return promote_point_to_top(model, X_1001)


        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_1001 on the given reference triangle."""
            model = create_model('X_1001 = MIDPOINT OF X_1 AND X_9')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01001(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-01001.json"))
            write_center_svg(
                model_acute,
                here / "x-01001.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1001¶

X₁₀₀₁¶