The First Six Books of the Elements of Euclid

by Euclid; John Casey

2. Since the triangle ABH is similar to CDE, we have [xix.].

In like manner,

	AHI : CEF in the duplicate ratio of AH : CE;
	hence	ABH : CDE	= AHI : CEF [V. xi.].
	Similarly,	AHI : CEF	= AIJ : CFG.

In these equal ratios, the triangles ABH, AHI, AIJ are the antecedents, and the triangles CDE, CEF, CFG the consequents, and [V. xii.] any one of these equal ratios is equal to the ratio of the sum of all the antecedents to the sum of all the consequents; therefore the triangle ABH : the triangle CDE :: the polygon ABHIJ : the polygon CDEFG.

3. The triangle ABH : CDE in the duplicate ratio of AB : CD [xix.]. Hence (2) the polygon ABHIJ : the polygon CDEFG in the duplicate ratio of AB : CD.

Cor. 1.—The perimeters of similar polygons are to one another in the ratio of their homologous sides.

Cor. 2.—As squares are similar polygons, therefore the duplicate ratio of two lines is equal to the ratio of the squares described on them (compare Annotations, V. Def. x.).

Cor. 3.—Similar portions of similar figures bear the same ratio to each other as the wholes of the figures.

Cor. 4.—Similar portions of the perimeters of similar figures are to each other in the ratio of the whole perimeters.