## Book VI, Proposition 21

Figures which are similar to the same rectilineal figure are also similar to one another.

Τὰ τῷ αὐτῷ εὐθυγράμμῳ ὅμοια καὶ ἀλλήλοις ἐστὶν ὅμοια. Ἔστω γὰρ ἑκάτερον τῶν Α, Β εὐθυγράμμων τῷ Γ ὅμοιον: λέγω, ὅτι καὶ τὸ Α τῷ Β ἐστιν ὅμοιον. Ἐπεὶ γὰρ ὅμοιόν ἐστι τὸ Α τῷ Γ, ἰσογώνιόν τέ ἐστιν αὐτῷ καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει. πάλιν, ἐπεὶ ὅμοιόν ἐστι τὸ Β τῷ Γ, ἰσογώνιόν τέ ἐστιν αὐτῷ καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει. ἑκάτερον ἄρα τῶν Α, Β τῷ Γ ἰσογώνιόν τέ ἐστι καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει [ ὥστε καὶ τὸ Α τῷ Β ἰσογώνιόν τέ ἐστι καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει ]. ὅμοιον ἄρα ἐστὶ τὸ Α τῷ Β: ὅπερ ἔδει δεῖξαι. | Figures which are similar to the same rectilineal figure are also similar to one another. For let each of the rectilineal figures A, B be similar to C; I say that A is also similar to B. For, since A is similar to C, it is equiangular with it and has the sides about the equal angles proportional. [VI. Def. 1] Again, since B is similar to C, it is equiangular with it and has the sides about the equal angles proportional. |