Kitap 6
20
Similarly we can prove, if BD, GK be joined, that, as the triangle BEC is to the triangle LGH, so also is the triangle ECD to the triangle LHK.
20
And since, as the triangle ABE is to the triangle FGL, so is EBC to LGH, and further ECD to LHK, therefore also, as one of the antecedents is to one of the consequents so are all the antecedents to all the consequents; [V. 12 therefore, as the triangle ABE is to the triangle FGL, so is the polygon ABCDE to the polygon FGHKL.
20
But the triangle ABE has to the triangle FGL a ratio duplicate of that which the corresponding side AB has to the corresponding side FG; for similar triangles are in the duplicate ratio of the corresponding sides. [VI. 19]
20
Therefore the polygon ABCDE also has to the polygon FGHKL a ratio duplicate of that which the corresponding side AB has to the corresponding side FG.
20
Therefore etc.
20
Porism. Similarly also it can be proved in the case of quadrilaterals that they are in the duplicate ratio of the corresponding sides. And it was also proved in the case of triangles; therefore also, generally, similar rectilineal figures are to one another in the duplicate ratio of the corresponding sides. Q. E. D.
PROPOSITION 21.
21
Figures which are similar to the same rectilineal figure are also similar to one another.
21
For let each of the rectilineal figures A, B be similar to C; I say that A is also similar to B.
21
For, since A is similar to C, it is equiangular with it and has the sides about the equal angles proportional. [VI. Def. 1]
21
Again, since B is similar to C, it is equiangular with it and has the sides about the equal angles proportional.
21
Therefore each of the figures A, B is equiangular with C and with C has the sides about the equal angles proportional; therefore A is similar to B. Q. E. D.
PROPOSITION 22.
