If in an equilateral and equiangular pentagon straight lines subtend two adjacent angles, they cut one another in extreme and mean ratio, and the greater segments are equal to the side of the pentagon.
Proof
Construct the pentagon inscribed in a circle (IV.11). Two diagonals
form an isosceles triangle with vertex angle 36∘ (I.32 / IV.10);
by similarity (VI.4) the diagonal-segment ratio matches the
extreme-and-mean ratio.