Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/225

Rh But, because the polygons are similar, [Hypothesis. therefore the whole angle ABC is equal to the whole angle FGH, [VI. Definition 1. therefore the remaining angle EBC is equal to the remain- ing angle LGH. [Axiom 3. And, because the triangles ABE and FGL are similar, therefore EB is to BA as LG is to GF; and also, because the polygons are similar, [Hypothesis. therefore AB is to BC as FG is to GH ; [VI. Definition 1. therefore, ex aequali, EB is to BC as LG is to GH; [V. 22. that is, the sides about the equal angles EBC and LGH are proportionals ; therefore the triangle EBC is equiangular to the triangle LGH; [VI. 6. and therefore these triangles are similar. [VI. 4. For the same reason the triangle ECD is similar to the triangle LHK. Therefore the similar polygons ABCDE, FGHKL may be divided into the same number of similar triangles.

Also these triangles shall have, each to each, the same ratio which the polygons have, the antecedents being ABE, EBC, ECD, and the consequents FGL, LGH, LHK; and the polygon ABCDE shall be to the polygon FGHKL in the duplicate ratio of AB to FG.

For, because the triangle ABE is similar to the tri- angle FGL, therefore ABE is to FGL in the duplicate ratio of EB to LG. [VI. 19. For the same reason the triangle EBC is to the triangle LGH in the duplicate ratio of EB to LG. Therefore the triangle ABE is to the triangle FGL as the triangle EBC is to the triangle LGH. [V. 11. Again, because the triangle EBC is similar to the tri- angle LGH, therefore EBC is to LGH in the duplicate ratio of EC to LH. [VI. 19.