Construct an n-gon (polygon with n sides) for which n given points serve as midpoints of its sides.
Below there is a Java applet that may help you gain insight into the problem. The applet can be in two modes: "Points" and "Play" as indicated on the right most button in the upper right corner of the applet. Clicking the button changes the mode.
If, for whatever reason, you decide to work with a different set of points - press the Reset button and start again.
A word of advice. First of all, start with a simple case. Try just a few points, say, 3, 4, 5. Secondly, attempt to visualize a polygon and then place the points approximately at the middle of its sides. Thirdly, pay attention to the blue circle.
By now you ought to have arrived at the following conclusions:
For odd n, the blue circle indicates one of the polygon's vertices. Once you move the cursor into the blue circle the problem seems to have been solved. Now, actually the applet provides an experimental device. It may help you to fathom a proof but it can't substitute for a rigorous demonstration.
Useful for the proof could be the following observation: Let there be given a segment AB and a point O. Consider a reflection A'B' of AB in the point O. In other words, in order to obtain A'B' rotate AB around O 180o. Now AB || A'B' but they have a different orientation. If you continue reflecting A'B' in another point O' to get A''B'' and so on. Then again after an odd number (1 in particular) of reflections the last segment will be parallel to AB but having a different orientation. See if you can now complete the proof.