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X₃₆₄₅¶

POINT ALKAID

Denote the incenter and excenters by I, I_A, I_B, I_C. Let J_A be the incenter of triangle BCI, and define J_B and J_C cyclically. The lines I_AJ_A, I_BJ_B, I_CJ_C concur in X₃₆₄₅.

X₃₆₄₅ (POINT ALKAID) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3645 = POINT ALKAID."""

        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,
            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_03645(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3645 = POINT ALKAID on reference triangle ABC."""
            X_3645 = eval_trilinear_point(model, A, B, C, '(1 + cos(B/2) + cos(C/2) - cos(A/2))', ID="X_3645", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_168 = eval_trilinear_point(model, A, B, C, 'sin(A) - sin(B) - sin(C) + 2*(cos(A/2) + sin(B/2 - A/2) + sin(C/2 - A/2))', ID="X_168", classes=["center", "incident"])
            central_line(model, X_1, X_168, 1, 168, incident=True)
            X_40 = eval_trilinear_point(model, A, B, C, 'cos(B) + cos(C) - cos(A) - 1', ID="X_40", classes=["center", "incident"])
            X_483 = eval_trilinear_point(model, A, B, C, '(sec(A/4))**(2)', ID="X_483", classes=["center", "incident"])
            central_line(model, X_40, X_483, 40, 483, incident=True)
            X_258 = eval_trilinear_point(model, A, B, C, '1/(cos(B/2) + cos(C/2) - cos(A/2))', ID="X_258", classes=["center", "incident"])
            X_1127 = eval_trilinear_point(model, A, B, C, '(sin(A/4))/sin(3*A/4)', ID="X_1127", classes=["center", "incident"])
            central_line(model, X_258, X_1127, 258, 1127, incident=True)
            return promote_point_to_top(model, X_3645)


        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_3645 on the given reference triangle."""
            model = create_model('X_3645 = POINT ALKAID')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03645(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-03645.json"))
            write_center_svg(
                model_acute,
                here / "x-03645.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3645¶

X₃₆₄₅¶