mjhjmtu

documentclassstandalone
usepackagetikz
usetikzlibraryintersections
usetikzlibrarycalc
begindocument

begintikzpicture

 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);

endtikzpicture
enddocument

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.

documentclassstandalone
usepackagetikz
usetikzlibraryintersections
usetikzlibrarycalc
begindocument

begintikzpicture

 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) ;

endtikzpicture
enddocument

As you see, the red lines are parallel.

One may want cast this into a style:

documentclassstandalone
usepackagetikz
usetikzlibraryintersections
usetikzlibrarycalc
begindocument
tikzsetline through/.style args=#1 parallel to line through #2 and #3 and
length #4insert path=%
let p1=($(#3)-(#2)$),n1=atan2(y1,x1) in (#1) -- ++ (n1:#4)
begintikzpicture

 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) ;

endtikzpicture
enddocument

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

documentclassarticle
usepackagetikz
usetikzlibraryintersections
usetikzlibrarycalc
begindocument

begintikzpicture

 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);

endtikzpicture
enddocument