Divide a Segment into n Equal Parts 

Parallel lines can be used to divide a segment into equal parts as the following example illustrates.

Given:  Segment AB, locate points P, Q, R, and S so AP = PQ = QR = RS = SB.

Solution:  Draw any angle less than 45^o with vertex A and copy that angle in the reverse direction at vertex B:

Then set the compass at any convenient length and mark off 4 equally-spaced points on the new sides of each angle, starting at each vertex. 

Finally, connect those points in order with lines that intersect AB at points P, Q, R, and S:

Here is an animation of this construction done slightly differently.

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