← etc / Encyclopedia of Triangle Centers

X₃₆₄₆¶

POINT ALMACH

Denote the incenter and excenters by I, I_A, I_B, I_C. Let K_A be the centroid of triangle BCI, and define K_B and K_C cyclically. The lines I_AK_A, I_BK_B, I_CK_C concur in X₃₆₄₆.

X₃₆₄₆ (POINT ALMACH) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3646 = POINT ALMACH."""

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


        def construct_x_03646(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3646 = POINT ALMACH on reference triangle ABC."""
            X_3646 = eval_trilinear_point(model, A, B, C, '(5 + cos(A) + (3 + cos(A))*(csc(A))*(sin(B) + sin(C)))', ID="X_3646", classes=["selected", "target"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_40 = eval_trilinear_point(model, A, B, C, 'cos(B) + cos(C) - cos(A) - 1', ID="X_40", classes=["center", "incident"])
            central_line(model, X_2, X_40, 2, 40, incident=True)
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_210 = eval_trilinear_point(model, A, B, C, '(b + c)*(b + c - a)', ID="X_210", classes=["center", "incident"])
            central_line(model, X_1, X_210, 1, 210, incident=True)
            X_57 = eval_trilinear_point(model, A, B, C, '1/(b + c - a)', ID="X_57", classes=["center", "incident"])
            X_191 = eval_trilinear_point(model, A, B, C, '(b + c - a)*(b*c + c*a + a*b) + b**(3) + c**(3) - a**(3)', ID="X_191", classes=["center", "incident"])
            central_line(model, X_57, X_191, 57, 191, incident=True)
            return promote_point_to_top(model, X_3646)


        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_3646 on the given reference triangle."""
            model = create_model('X_3646 = POINT ALMACH')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03646(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-03646.json"))
            write_center_svg(
                model_acute,
                here / "x-03646.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3646¶

X₃₆₄₆¶