[Tex/LaTex] How to define a new triangle symbol with middle line for use as a maths operator

Question

I would like to use a specific symbol, a triangle with a middle line, as a mathematical operator. Here is roughly how the symbol ought to look like:

[This was generated with tikz by appropriating the answer by Heiko Oberdiek to the question Harry Potter symbols].

My intended usage has nothing to do with Harry Potter, instead I would like to use it as a unary mathematical operator that behaves well with subscripts, for instance.

Intended syntax

Ideally, I would define a command \triangleline which gives the symbol along with some code that ensures it behaves well in common mathematical environments. After this, I would define \trilineop

\newcommand{\trilineop}[1]{\operatorname{\triangleline_\mathrm{e}}{\left(#1\right)}}

for use in my document.

But I am not sure how to go about this, whether this is a good plan, or what would be a better plan.
Maybe there's a neat way to build this symbol out of \triangle somehow?

(Use of \Delta in undesired because the triangle sides are asymmetric in many fonts).

---

Here is an example environment which might help as a starting point.

\documentclass{beamer}

\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{tikz}

%Code here to define \triangleline
%%
%%

\newcommand{\trilineop}[1]{\operatorname{\triangleline_\mathrm{e}}{\left(#1\right)}}

\begin{document}
    
    \begin{frame}{}
      $$\trilineop{\boldsymbol{A}}$$
    \end{frame}
  
\end{document}

Here is the tikz code mentioned earlier used to generate the example image. (I used this as a black box).

\begin{tikzpicture}[baseline=0]
    \def\a{1cm}
    \pgfmathsetlengthmacro\radius{\a/2 * tan(30)}
    \draw[thick]
    (0, 0) -- (60:\a) -- (\a, 0) -- cycle
    (\a/2, 0) -- (60:\a)
    ;
\end{tikzpicture}

Answer 1

I am not a big fan of using TikZ everywhere so I propose here a more minimal version based on good old \bigtriangleup. I also am not a big fan of \left/\right at all costs, so I propose a definition based on \DeclarePairedDelimiterXPP from mathtools. The automatic scaling can be obtained with the starred version, while manual sizes can be passed as optional parameter.

\documentclass{article}

\makeatletter
\newcommand*{\triangleline}{\mathpalette\@triangleline\relax}
\newcommand*{\@triangleline}[2]{{%
    \sbox0{\m@th$#1\bigtriangleup$}%
    \dimen@\fontdimen8
       \ifx#1\displaystyle\textfont\else
       \ifx#1\textstyle\textfont\else
       \ifx#1\scriptstyle\scriptfont\else
       \scriptscriptfont\fi\fi\fi 3
   \ooalign{\hfil\hbox{\vrule\@width\dimen@\@height\ht0\@depth-.5\dimen@}\hfil\cr\box0}%
}}
\makeatother

\usepackage{mathtools}% for the following
\DeclarePairedDelimiterXPP{\trilineop}[1]{\mathop{\triangleline_{\mathrm{e}}}}{(}{)}{}{#1}

\begin{document}

\parskip=\bigskipamount

$\triangleline_{\triangleline_{\triangleline}}$

$2\trilineop{x}$

$2\trilineop[\big]{\frac{x}{2}}$

$2\trilineop*{\dfrac{x}{2}}$

\end{document}