While tidying my office last year I came across a book of Entrance Examinations for Hertford College, Oxford dated 1926. The first exam paper was on Geometry and began with this:

Construct a pentagon given the midpoints of its five sides.

Can you do it?

I wonder if the methods you use are the same as the 1926 students used?

Here is an extension to the same problem. Given an odd number of points in a plane, explain how you would construct a polygon (using only ruler and compasses) with the given points as the midpoints of the sides. Find a necessary and sufficient condition to do the construction for an even number of points. You may like to compare this to the Polycircles problem in the February 2000 15 + Challenges.

If you have a java enabled browser you can experiment with the interactive diagram below by clicking and dragging the green points