Construction of a regular pentagon.

It is well-known that a regular pentagon is constructible. Here is the simplest way that I know of.

Note that in a regular pentagon inscribed in a unit circle shown on the right, the angle `angle BAC = 36^o`. This makes the length `AB = (1+sqrt(5))/2`, as `triangle ABC` is a right triangle.

So all we need is to be able to construct the length `(1+sqrt(5))/2`. To do so, observe the diagram on the right, where two smaller circle of radius `1/2` is inscribed in a unit circle. If we extend the segment of `AP` to `AQ`, then we will find that `AQ = (1+sqrt(5))/2`, precisely what we need!