The target shape for this puzzle is a regular pentagon.  The Greeks of Euclid's time knew how to inscribe a regular pentagon and a regular triangle inside a circle, but for the next 2000 years no one was able to add to these discoveries.  Then in 1801 Karl Friedrich Gauss proved that: 

A regular polygon can be constructed with compass and straight-edge when and only when the number of its sides has the form 2^m*p1*p2...pv, where p1, p2, ..., pv are all different prime numbers of the form 2^n + 1

(See "100 Great Problems of Elementary Mathematics; Their History and Solution" by Heinrich Dörrie for a very lucid proof.)

For m = 0, v = 1 and p1 = 3 and p1 = 5 we obtain the cases of the regular triangle and regular pentagon, respectively.  Since p1 = 17 is a prime of the form 2^n + 1 Gauss's result says that a regular Heptadecagon can also be constructed with compass and straight-edge.

Gauss was one of the world's greatest mathematicians and he made many outstanding discoveries but he was particularily proud of this result, so much so that he gave instructions that a regular Heptadecagon should be inscribed on his grave.