Me!
Well, the constructions for the pentagon are not difficult to explain, but not so immediate to prove. I was used to prove one of them in a third year class of university course in mathematics, and it required to master complex numbers ;indeed, the key fact here is that the vertices of the regular polygon with n edges can be seen as the n-roots of 1, i.e.they correspond to complex numbers z= (a + b i) such that z times z times z ... n times gives 1. Here you should know the i times i gives -1.
The important fact here is to undestand that we are talking about "precise" constructions made just with ruler and compass.
There is a famous theorem of Gauss stating that just few polygons are constructible that way - I remember 3,5, 17,255... there is a rule...
the other construction you can find are approximate, in the sense that the error is so small that they are good for applications (for example, building gears)
