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".
To understand the Third Theorem, compare the Copernican diagram shown on page 39 of the Harvard copy of De Revolutionibus to that for the derivation of sin(A-B) found in the above cut-the-knot web page
In De Revolutionibus Orbium Coelestium, Copernicus does not refer to Pythagoras's theorem by name but uses the term 'Porism' – a word which in this particular context would appear to denote an observation on – or obvious consequence of – another existing theorem. The 'Porism' can be viewed on pages 36 and 37 of DROC (Harvard electronic copy)
To understand the Third Theorem, compare the Copernican diagram shown on page 39 of the Harvard copy of De Revolutionibus to that for the derivation of sin(A-B) found in the above cut-the-knot web page
