Path transformations cause intersections to be computed incorrectly

As of 3.1.6, path intersections are computed incorrectly if path coordinates are e.g., scaled. The regression was likely caused by the fix for issue #889 in commit 908db0011cc96b4092b45de7d432e3ead32f4323.

Minimal working example (MWE)

\documentclass[tikz]{standalone}

\usetikzlibrary{calc}
\usetikzlibrary{intersections}

\begin{document}

\begin{tikzpicture}
[
  % NOTE Comment the following line to obtain the expected result
  every path/.style={scale=1.25},
  declare function={
    angle=22.5;
  },
]
  \draw[densely dotted] (0,0) -- +(angle:0.5) coordinate (A);
  \draw (A)
    --
    %node[sloped,below] {\(l_1\)}
    +(angle:1.2)
    coordinate (B)
    ;

  \draw
    ($(B)+(2.25,0)$) coordinate (C)
    --
    +(-22.5:1) coordinate (D)
    ;
  \draw[densely dotted] (D) -- +(-angle:.5) coordinate (E);

  \path[name path=BC] (B) -- (angle:5) coordinate (AB ext);
  \path[name path=CD] (C) -- +(-angle:-3) coordinate (CD ext);

  \path[name intersections={of=BC and CD,by=inter}];

  % The intersection of extended lines
  \draw[mark=x,very thick,scale=1.5] plot coordinates {(inter)};
\end{tikzpicture}

\end{document}

\end{document}

mwe-1

Without scaling, the intersection given by the cross is placed correctly:

mwe-after-1

pgf-tikz / pgf

Path transformations cause intersections to be computed incorrectly #929