Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/224

200 straight lines be proportionals, as the first is to the third, so is any triangle described on the first to a similar and similarly described triangle on the second.

PROPOSITION 20. THEOREM.

Similar polygons may he divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides.

Let ABCDE, FGHKL be similar polygons, and let AB be the side homologous to the side FG: the polygons ABCDE, FGHKL may be divided into the same number of similar triangles, of which each shall have to each the same ratio which the polygons have; and the polygon ABCDE shall be to the polygon FGHKL in the duplicate ratio of AB to FG.

Join BE, EC, GL, LH. Then, because the polygon ABCDE is similar to the polygon FGHKL, [Hypothesis. the angle BAE is equal to the angle GFL, and BA is to AE as GF is to FL. [VI. Definition 1. And, because the triangles ABE and FGL have one angle of the one equal to one angle of the other, and the sides

about these equal angles proportionals, therefore the triangle ABE is equiangular to the triangle FGL, [VI. 6. and therefore these triangles are similar; [VI. 4. therefore the angle ABE is equal to the angle FGL.