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.
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.
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.
Quoniam enim simile est A ei quod est G,
et equiangulum est ipsi et circa equales angulos latera proportionalia habet.
(Rursum quoniam simile est B ei quod est G,
et equiangulum est ipsi et circa equales angulos latera proportionalia habet.)
Utrumque eorum que sunt A, B ei quod est G et equiangulum est et circa equales angulos latera proportionalia habet. Quare et A ei quod est B et equiangulum est et circa equales angulos latera proportionalia habet.
Simile ergo B ei quod est A.
Quod oportet ostendere.
Complete text
Book I