Wie mich @murmeltier aufklärte brauchte braucht man eine Definition der Koordinatenachsen vom Typ:
\begin{tikzpicture}[...,
x={({cos(16.5)*1cm}, {-sin(16.5)*1cm})},
y={({0.9*cos(10)*1cm}, {0.9*sin(10)*1cm})},
z={(0cm,1cm)},
]
Damit
[![alt text][1]][1]
\documentclass[margin=5pt, tikz]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\pgfmathsetmacro{\a}{0.85}
\pgfmathsetmacro{\m}{\a*sqrt(3)/2}
\pgfmathsetmacro{\H}{sqrt(1-\m^2/4)} % Höhe
\pgfmathsetmacro{\h}{sqrt(\H^2-\m^2/4)} % Mantellinie
\begin{tikzpicture}[scale=5.7,
font=\footnotesize,
x={({cos(16.5)*1cm}, {-sin(16.5)*1cm})},
y={({0.9*cos(10)*1cm}, {0.9*sin(10)*1cm})},
z={(0cm,1cm)},
]
\coordinate[label=left:$A$] (A) at (0,0,0);
\coordinate[label=below:$B$] (B) at (\a,0,0);
\coordinate[label=right:$C$] (C) at (0,\a,0);
%\coordinate[label=below:$D$] (D) at (0,0,\a);
\draw[fill=lightgray!50] (A) -- (B) -- (C) --cycle;
\path[] (A) -- (B) node[below, near start]{$a$};
\path[] (B) -- (C) node[right, near start]{$a$};
\path[] (A) -- (C) node[above, near start]{$a$};
\path[] (A) -- ($(B)!0.5!(C)$) coordinate(Ha);
\path[] (B) -- ($(A)!0.5!(C)$) coordinate(Hb);
\coordinate[] (M) at (intersection of A--Ha and B--Hb);
\pgfmathsetmacro{\k}{0.05*\m}
\draw[] ($(M)!-\k cm!(C)$) -- ($(M)!\k cm!(C)$);
\draw[] ($(M)!-\k cm!(B)$) -- ($(M)!\k cm!(B)$);
\path[] (M) --+ (0,0,\H) coordinate[label=$S$] (S);
\draw[] (A) -- (S) node[midway, left]{$1$};
\draw[] (B) -- (S) node[midway, right]{$1$};
\draw[] (C) -- (S) node[midway, right]{$1$};
\draw[densely dashed] (S) -- (M) node[near end, right]{$H$};
\draw[] (S) -- ($(A)!0.5!(B)$) node[midway, left]{$h$} coordinate(Mc);
\pgfmathsetmacro{\b}{0.3*\a}
\begin{scope}[-latex, shift={(0,0,\H-\b)}
]
\foreach \P/\s/\Pos in {(\b,0,0)/x/below, (0,\b,0)/y/above, (0,0,\b)/z/right}
\draw[] (0,0,0) -- \P node[\Pos, pos=0.9,inner sep=3pt]{$\s$};
\end{scope}
% Punkte
\foreach \P in {A,B,C,Mc,S}
\draw[fill=black!1, scale=1/5.7] (\P) circle (1.75pt);
%\node[anchor=north west, align=left, yshift=-1cm, draw] at (A){
%a = \a cm \\
%m = \m cm \\
%H = \H cm \\
%h = \h cm
%};
\end{tikzpicture}
\end{document}
[1]: https://texwelt.de/wissen/upfiles/55555555_405.png
