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Euclid: Elementa

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κα῾ 
PROPOSITION 21. 
21 
 
 
第二十一題 
Τὰ τῷ αὐτῷ εὐθυγράμμῳ ὅμοια καὶ ἀλλήλοις ἐστὶν ὅμοια. 
Figures which are similar to the same rectilineal figure are also similar to one another. 
Eidem rectilineo similia et sibi invicem sunt similia. 
 
 
兩直線形。各與他直線形相似。則自相似。 
Ἔστω γὰρ ἑκάτερον τῶν Α, Β εὐθυγράμμων τῷ Γ ὅμοιον:  λέγω, ὅτι καὶ τὸ Α τῷ Β ἐστιν ὅμοιον. 
For let each of the rectilineal figures A, B be similar to C;  I say that A is also similar to B. 
Esto enim utrumque rectilineorum A, B ei quod est G simile.  Dico quoniam et ei quod est A, B est simile. 
   
   
   
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. 
             
             
             
 
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