Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/227

Rh But any rectilineal figure described on 'AB' is to the similar and similarly described rectilineal figure on FG in the duplicate ratio of AB to FG, [Corollary 1. Therefore as AB is to M, so is the figure on AB to the figure on FG; [V. 11. and this was shewn before for triangles. [VI. 19, Corollary. Wherefore, universally, if three straight lines be propor-tionals, as the first is to the third, so is any rectilineal figure described on the first to a similar and similarly described rectilineal figure on the second.

PROPOSITION 21. THEOREM.

Rectilineal figures which are similar to the same rectilineal figure, are also similar to each other.

Let each of the rectilineal figures A and B be similar to the rectilineal figure C: the figure A shall be similar to the figure B.

For, because A is similar to C, [Hyp. A is equiangular to C, and A and C have their sides about the equal angles proportionals. [VI. Def. 1. Again, because B is similar to C, [Hyp. B is equiangular to C, and B and C have their sides about the equal angles proportionals. [VI. Definition 1. Therefore the figures A and B are each of them equiangular to C, and have the sides about the equal angles of each of them and of C proportionals. Therefore A is equiangular to B, [Axiom 1. and A and B have their sides about the equal angles proportionals; [V. 11 therefore the figure A is similar to the figure B. [VI. Def. 1.

Wherefore, rectilineal figures &c.