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.