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X₁₃¶

1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA', BB', CC' concur in X₁₃. If each of the angles A, B, C is < 2π/3, then X₁₃ is the only center X for which the angles BXC, CXA, AXB are equal, and X₁₃ minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X₁₃ is equilateral.

X₁₃ (1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_13 = 1st ISOGONIC CENTER."""

        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,
            outer_equilateral_triangles,
            perspector,
            promote_point_to_top,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00013(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_13 = 1st ISOGONIC CENTER on reference triangle ABC."""
            Ea, Eb, Ec = outer_equilateral_triangles(model, A, B, C)
            X_13 = perspector(model, A, B, C, Ea, Eb, Ec, ID="X_13", classes=["selected", "target"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_16 = eval_trilinear_point(model, A, B, C, 'sin(A-  pi  /3)', ID="X_16", classes=["center", "incident"])
            central_line(model, X_2, X_16, 2, 16, incident=True)
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", 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_3, X_17, 3, 17, incident=True)
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_14 = eval_trilinear_point(model, A, B, C, 'csc(A -  pi  /3)', ID="X_14", classes=["center", "incident"])
            central_line(model, X_6, X_14, 6, 14, incident=True)
            return promote_point_to_top(model, X_13)


        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_13 on the given reference triangle."""
            model = create_model('X_13 = 1st ISOGONIC CENTER')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00013(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-00013.json"))
            write_center_svg(
                model_acute,
                here / "x-00013.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X13¶

X₁₃¶