TikZ : intersections

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\documentclass[margin = 2 mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
% les points
\node (A) at (1, 0) {A} ;
\node (B) at (2, 0) {B} ;
\node (C) at (1, 3) {C} ;
\node (D) at (2, 3) {D} ;
\node (p) at (0, 1.6) {p} ;
\node (q) at (3, 1.4) {q} ;
% je nomme les segments de droite
\draw [name path = AC] (A) -- (C) ;
\draw [name path = BD] (B) -- (D) ;
\path [name path = pq] (p) -- (q) ;
% je nomme E et F les intersections entre le segment de droite pq
% et les lignes verticales AC et BD
\path [name intersections = {of = AC and pq, by = E}] ;
\path [name intersections = {of = BD and pq, by = F}] ;
% segment pE dessiné en trait plein
\draw[red] (p) -- (E) ;
% segment EF en pointillés
\draw[red, densely dotted] (E) -- (F) ;
% segment Fq en trait plein
\draw[red] (F) -- (q) ;
\end{tikzpicture}
\end{document}
 
 
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