Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio

(Proposition 8 from Book 13 of Euclid's “Elements”)

If straight lines subtend two consecutive angles of an equilateral and equiangular pentagon then they cut one another in extreme and mean ratio, and their greater pieces are equal to the sides of the pentagon. * For let the two straight lines, $AC$ and $BE$, cutting one another at point $H$, have subtended two consecutive angles, at $A$ and $B$ (respectively), of the equilateral and equiangular pentagon $ABCDE$. * I say that each of them has been cut in extreme and mean ratio at point $H$, and that their greater pieces are equal to the sides of the pentagon.