There are different ways you could crack this Tough Nut but if you have one good idea about it do let us hear from you.

Vassil Vassilev, Y10, Lawnswood High School, has an idea for this construction based on nested pentagons, and 5 pointed stars within them, which are all enlargements of each other. You might like to play with this idea. There are several different construction methods and we'll publish alternatives if you send them in.

To start on a different approach you might like first to think of the same problem for a triangle where you are given the midpoints of the sides and have to find the vertices. Try doing this with coordinates. You can choose any set of 3 midpoints, write down and solve some simultaneous equations and find the vertices. You can then extend the method to pentagons. Then you could consider the differences between cases of polygons with an odd number of sides and polygons with an even number of sides.

Finally you can compare this problem to Polycircles in The February 2000 15+ Challenges.