Tukukkk

documentclass{standalone}

usepackage{tikz}

usetikzlibrary{intersections}

usetikzlibrary{calc}

begin{document}



begin{tikzpicture}



    coordinate (A) at (0,0);

    coordinate (B) at (2,4);

    coordinate (C) at (8,0);

    coordinate (M) at (4,0);



    path[name path=Circle] (B) circle [radius=3cm];

    path[name path=AB] (A)--(B);

    path[name path=BC] (B)--(C);

    path [name intersections={of=Circle and BC}];

    coordinate (E) at (intersection-1);

    path [name intersections={of=Circle and AB}];

    coordinate (D) at (intersection-1);



    draw[thick](A)--(B)--(C)--cycle;

    draw[thick](B) circle [radius=3cm];

    draw[thick](D)--(E)  (M)--(B);



    % From a point draw a parallel line by calculating the vector

    coordinate (P) at (intersection cs:first line={(B)--(C)}, second line={(A) -- +($(E)-(D)$)});



    % Here is my problem:

    coordinate (N) at (intersection cs:first line={(B)--(M)}, second line={(C) -- +($(D)-(E)$)});



    draw[thick, red] (A)--(P) (C)--(N);



end{tikzpicture}

end{document}

Question: The second drawn line, which is supposed to also be a parallel, is not. What am I doing wrong?

If you use the + syntax in this context, it seems not to always give you what one may expect. In general, + means "relative to the first coordinate of this path". But who knows what the first coordinate is in the context of intersection cs:? It is, however, not too difficult to produce the parallel lines just by computing the slope and using it. For the sake of clarity, I labeled all coordinates.

documentclass{standalone}

usepackage{tikz}

usetikzlibrary{intersections}

usetikzlibrary{calc}

begin{document}



begin{tikzpicture}



    coordinate[label=left:$A$] (A) at (0,0);

    coordinate[label=above:$B$] (B) at (2,4);

    coordinate[label=right:$C$] (C) at (8,0);

    coordinate[label=45:$M$] (M) at (4,0);



    draw[name path=Circle] (B) circle [radius=3cm];

    path[name path=AB] (A)--(B);

    path[name path=BC] (B)--(C);

    path [name intersections={of=Circle and BC}];

    coordinate[label=right:$E$] (E) at (intersection-1);

    path [name intersections={of=Circle and AB}];

    coordinate[label=left:$D$] (D) at (intersection-1);



    draw[thick](A)--(B)--(C)--cycle;

    %draw[thick](B) circle [radius=3cm];

    draw[thick](D)--(E)  (M)--(B);



    % From a point draw a parallel line by calculating the vector

    path  let p1=($(E)-(D)$),n1={atan2(y1,x1)} in 

    ($(A)+(n1:1)$) coordinate (auxA) 

    ($(C)+(n1:1)$) coordinate (auxC);

    coordinate[label=right:$P$] (P) at (intersection cs:first line={(B)--(C)},

    second line={(A) -- (auxA)});



    % Here is my problem:

    coordinate[label=below:$N$] 

    (N) at (intersection cs:first line={(B)--(M)},

    second line={(C) -- (auxC)});



    draw[thick, red] (A)--(P) (C)--(N) ;



end{tikzpicture}

end{document}

As you see, the red lines are parallel.

One may want cast this into a style:



documentclass{standalone}

usepackage{tikz}

usetikzlibrary{intersections}

usetikzlibrary{calc}

begin{document}

tikzset{line through/.style args={#1 parallel to line through #2 and #3 and

length #4}{insert path={%

let p1=($(#3)-(#2)$),n1={atan2(y1,x1)} in (#1) -- ++ (n1:#4)}}}

begin{tikzpicture}



    coordinate[label=left:$A$] (A) at (0,0);

    coordinate[label=above:$B$] (B) at (2,4);

    coordinate[label=right:$C$] (C) at (8,0);

    coordinate[label=45:$M$] (M) at (4,0);



    draw[name path=Circle] (B) circle [radius=3cm];

    path[name path=AB] (A)--(B);

    path[name path=BC] (B)--(C);

    path [name intersections={of=Circle and BC}];

    coordinate[label=right:$E$] (E) at (intersection-1);

    path [name intersections={of=Circle and AB}];

    coordinate[label=left:$D$] (D) at (intersection-1);



    draw[thick](A)--(B)--(C)--cycle;

    %draw[thick](B) circle [radius=3cm];

    draw[thick](D)--(E)  (M)--(B);



    % From a point draw a parallel line by calculating the vector

    path[line through=A parallel to line through E and D and length 1]

    coordinate (auxA)

    [line through=C parallel to line through E and D and length 1] 

    coordinate (auxC);



    coordinate[label=right:$P$] (P) at (intersection cs:first line={(B)--(C)},

    second line={(A) -- (auxA)});



    % Here is my problem:

    coordinate[label=below:$N$] 

    (N) at (intersection cs:first line={(B)--(M)},

    second line={(C) -- (auxC)});



    draw[thick, red] (A)--(P) (C)--(N) ;



end{tikzpicture}

end{document}

Use second line={(C)--($(C)+(D)-(E)$)}). The first one works as expected because (A)=(0,0)

documentclass{article}

usepackage{tikz}

usetikzlibrary{intersections}

usetikzlibrary{calc}

begin{document}



begin{tikzpicture}



    coordinate[label=A] (A) at (0,0);

    coordinate[label=B](B) at (2,4);

    coordinate[label=C] (C) at (8,0);

    coordinate[label=M] (M) at (4,0);



    path[name path=Circle] (B) circle [radius=3cm];

    path[name path=AB] (A)--(B);

    path[name path=BC] (B)--(C);

    path [name intersections={of=Circle and BC}];

    coordinate[label=E] (E) at (intersection-1);

    path [name intersections={of=Circle and AB}];

    coordinate[label=D] (D) at (intersection-1);



    draw[thick](A)--(B)--(C)--cycle;

    draw[thick](B) circle [radius=3cm];

    draw[thick](D)--(E)  (M)--(B);



    % From a point draw a parallel line by calculating the vector

    coordinate[label=P] (P) at (intersection cs:first line={(B)--(C)}, second line={(A) -- +($(E)-(D)$)});



    % Here is my problem:

    coordinate[label=N] (N) at (intersection cs:first line={(B)--(M)}, second line={(C)--($(C)+(D)-(E)$)}); % <------





    draw[thick, red] (A)--(P) (C)--(N);



end{tikzpicture}

end{document}