Given a line segment, a line parallel to it, and a straightedge, divide the segment into $n$ equal segments 
Given a line segment, a line parallel to it, and a straightedge, how to divide the segment into $n$ equal segments?

(With a straightedge, you are allowed only to draw straight lines. You are not allowed to mark off distances on the straightedge.)
 A: Partial Answer
Here's a solution for the case $n = 2$, which might provide you some inspiration for the more general case. The picture tells the whole story. The red line $PQ$ is the initial segment, the blue line is the parallel one. We pick points $A$ and $B$ on the blue line at random. We join each of $A$ and $B$ to each end of the initial segment. The lines $PA$ and $QB$ intersect at the magenta point; the lines $PB$ and $QA$ intersect at the blue point, and joining the blue and magenta points and taking the intersection with the original line $PQ$ gives us the red point, which is the midpoint of $PQ$. 
Note that I did need to assume I could pick two distinct points on a given line; I'm not certain that this is allowed in straightedge-only constructions. 
Clearly this solution generalizes to handle all cases where $n$ is a power of $2$ (just apply recursively to sub-lines), but I don't see how to do $n = 3$; presumably once that's clear, the rest is downwind sailing. 

A: Hint.  Assuming you are allowed a compass...
It might seem hard to divide the given line $n$ times but you can take the parallel line; extend it $n$ times to get a parallel line that is $n$ times longer than it was originally and you have successfully divided that big new line in $n$ equal parts.  
Is there any way to take those $n$ parts an the parallel line to make $n$ equal parts of the original line?
What do you know about parallel lines, and equal angles, and similar triangles?  
Time to brainstorm.... 
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