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X₁₇₅¶

ISOPERIMETRIC POINT

A point X is defined as an isoperimetric point of triangle ABC if |XB| + |XC| + |BC| = |XC| + |XA| + |CA| = |XA| + |XB| + |AB|. Veldkamp established that X = X₁₇₅, uniquely, for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient. See also X₁₇₆ and the 1st and 2nd Eppstein points, X₄₈₁, X₄₈₂.

X₁₇₅ (ISOPERIMETRIC POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_175 = ISOPERIMETRIC POINT."""

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


        def construct_x_00175(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_175 = ISOPERIMETRIC POINT on reference triangle ABC."""
            X_175 = eval_trilinear_point(model, A, B, C, '-1 + sec(A/2)*cos(B/2)*cos(C/2)', ID="X_175", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            central_line(model, X_1, X_7, 1, 7, incident=True)
            X_174 = eval_trilinear_point(model, A, B, C, 'sec(A/2)', ID="X_174", 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_174, X_483, 174, 483, incident=True)
            X_490 = eval_trilinear_point(model, A, B, C, '((cos(A))*(1 + csc(A)*sin(B)*sin(C)) - cos(B)*cos(C))', ID="X_490", classes=["center", "incident"])
            X_664 = eval_trilinear_point(model, A, B, C, '1/(a*(b - c)*(b + c - a))', ID="X_664", classes=["center", "incident"])
            central_line(model, X_490, X_664, 490, 664, incident=True)
            return promote_point_to_top(model, X_175)


        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_175 on the given reference triangle."""
            model = create_model('X_175 = ISOPERIMETRIC POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00175(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-00175.json"))
            write_center_svg(
                model_acute,
                here / "x-00175.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X175¶

X₁₇₅¶