Proposition 8 in Book XIII of Euclid's Elements proves by similar triangles the same result: namely that length a (the side of the pentagon) divides length b (joining alternate vertices of the pentagon) in "mean and extreme ratio".
And in analogous fashion Proposition 9 in Book XIII of Euclid's Elements proves by similar triangles that length c (the side of the decagon) divides the radius in "mean and extreme ratio".
cut-the-knot.org
An interesting article on the construction of a regular pentagon and determination of side length can be found at the following reference [1]
