Similar triangles are to one another in the duplicate ratio of the corresponding sides.
Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side. So, if we have two shapes somehow related to a line segment, their areas change proportionally to the square of the line segment. Although the statement itself is more general, the identity it leads to is obviously equivalent to that asserted in a more traditional version of the Pythagoras' Theorem. Thus, it could be said that, if the identity were to be shown in one particular case of similar shapes, it would follow for other shapes as well. But one case is actually trivial.