X₂

CENTROID

X₂ is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X₂ to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X₂ and also balances atop any knife-edge that passes through X₂. The triangles BXC, CXA, AXB have equal areas if and only if X = X₂.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₂ (CENTROID) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

"""GEOMETOR construction model for X_2 = CENTROID."""

        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,
            gergonne_point_of,
            incenter_of,
            mittenpunkt_of,
            nagel_point_of,
            orthocenter_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00002(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_2 = CENTROID on reference triangle ABC."""
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["selected", "target"], draw_medians=True)
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center"])
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center"])
            central_line(model, X_3, X_4, 3, 4)
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_8 = nagel_point_of(model, A, B, C, ID="X_8", classes=["center", "incident"])
            central_line(model, X_1, X_8, 1, 8, incident=True)
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            X_9 = mittenpunkt_of(model, A, B, C, ID="X_9", classes=["center", "incident"])
            central_line(model, X_7, X_9, 7, 9, incident=True)
            return promote_point_to_top(model, X_2)


        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_2 on the given reference triangle."""
            model = create_model('X_2 = CENTROID')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00002(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-00002.json"))
            write_center_svg(
                model_acute,
                here / "x-00002.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )