TikZ coordinate calculation accuracy

Question

Consider the following MWE:

\documentclass[border=5pt,tikz]{standalone}

\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}[join=bevel]
   \draw (0,0) coordinate (B) -- (6,0) coordinate (C)  
        -- (6,4.5) coordinate (A) -- cycle;
   \coordinate (F) at ($(A)!(C)!(B)$);
   \draw (C) -- (B) -- (0,8) coordinate (D)  -- cycle;
   \draw[fill=gray] (D) -- (A) -- (F) -- cycle;
\end{tikzpicture}

\end{document}

It yields

However, the corner of the gray triangle should be exactly on the line from the top left to the lower right corner. (This can be easily verified with GeoGebra or by calculation.)

Why does this happen?

Answer 1

As pointed out by Mark Wibrow in this answer the definition of \pgfpointnormalised can be corrected to obtain better precision.

\documentclass[border=5pt,tikz]{standalone}

\usepackage{tikz}
\usetikzlibrary{calc, spy}

% use the Mark Wibrow's correction
\makeatletter
\def\pgfpointnormalised#1{%
  \pgf@process{#1}%
  \pgfmathatantwo{\the\pgf@y}{\the\pgf@x}%
  \let\pgf@tmp=\pgfmathresult%
  \pgfmathcos@{\pgf@tmp}\pgf@x=\pgfmathresult pt\relax%
  \pgfmathsin@{\pgf@tmp}\pgf@y=\pgfmathresult pt\relax%
}
\makeatother

\begin{document}

\begin{tikzpicture}[join=bevel, spy using outlines={circle, magnification=7, size=17mm, connect spies}]
   \draw (0,0) coordinate (B) -- (6,0) coordinate (C)
        -- (6,4.5) coordinate (A) -- cycle;
   \coordinate (F) at ($(A)!(C)!(B)$);
   \draw (C) -- (B) -- (0,8) coordinate (D)  -- cycle;
   \draw[fill=gray] (D) -- (A) -- (F) -- cycle;

   \spy[red] on (F) in node at (1,2);
\end{tikzpicture}

\end{document}

Note : This is a copy of this answer.