A second solution is not so obvious.   It involves starting approximately 115.92 kilometers from the South Pole.  (The exact distance isn't important to understanding the solution, but is provided only for completeness.)  The starting location is shown on the map as a red dot.  The South Pole is in the center of the diagram. 

From the starting point, traveling 100 kilometers due south will take you to precisely 15.92 kilometers from the South Pole.   Next, when you travel 100 kilometers east, you will travel in a complete loop.   (A circle of radius 15.92 has a circumference of 100.)  Thus, when you travel 100 kilometers back north, you're returning over your first leg.

If you're astute, you'll notice that this second solution can start at any point on a circle that is 115.92 kilometers from the South Pole.  So, there are many places on the earth that satisfy the problem.   But these aren't the only locations that will work.  Can you find yet even another location on the earth where the navigational requirements are met?

That's what I love about this problem - what appears to be a simple solution quickly cascades into complexity.

A Third Solution