Page:The Works of Archimedes.djvu/510

318 Therefore, in the triangles EDA, ODM,

∠EDA = ∠ODM,

∠EAD = ∠OMD,

and the sides AD, MD are equal.

Hence the triangles are equal in all respects, and

Therefore

Moreover DE = DO; and it follows that, since DE is equal to the side of an inscribed hexagon, and DC is the side of an inscribed decagon, EC is divided at D in extreme and mean ratio [i.e. EC : ED = ED : DC]; "and this is proved in the book of the Elements." [Eucl. 9, "If the side of the hexagon and the side of the decagon inscribed in the same circle be put together, the whole straight line is divided in extreme and mean ratio, and the greater segment is the side of the hexagon."]