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Euclid: Elementa

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ΚΖʹ 
Proposition 27. 
27 
الشكل السابع والعشرون من المقالة الاولى 
atha saptaviṃśatitamaṃ kṣetram 
第二十七題 
Ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐναλλὰξ γωνίας ἴσας ἀλλήλαις ποιῇ, παράλληλοι ἔσονται ἀλλήλαις αἱ εὐθεῖαι. 
If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. 
Si in duas rectas recta incidens eos qui permutatim angulos equales alternis facit, equidistantes alternis erunt ille recte. 
اذا وقع خط (ع) مستقيم على خطين مستقيمين فصير الزاويتين المتبادلتين متساويتين فان الخطين (ط) متوازيان 
tatra rekhādvayor anyarekāyāṃ saṃpātaḥ kṛtaḥ tatraikakoṇo dvitīyadiksaṃbandhikoṇaś caitau tulyau yadi bhavataḥ tadā rekhādvayaṃ samānāntarālakaṃ bhavati | 
兩直線。有他直線交加其上若內相對兩角等。卽兩直線必平行。 
Εἰς γὰρ δύο εὐθείας τὰς ΑΒ, ΓΔ εὐθεῖα ἐμπίπτουσα ἡ ΕΖ τὰς ἐναλλὰξ γωνίας τὰς ὑπὸ ΑΕΖ, ΕΖΔ ἴσας ἀλλήλαις ποιείτω:  λέγω, ὅτι παράλληλός ἐστιν ἡ ΑΒ τῇ ΓΔ. 
For let the straight line EF falling on the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another;  I say that AB is parallel to CD. 
In duas enim rectas AB et GD recta incidens EZ eos qui permutatim angulos qui sint AEZ et EZD equales alternis faciat.  Dico quoniam equidistans est recta AB recte GD. 
مثاله ان خط (ه ز) وقع على خطى (ا ب) (ج د) فصير زاويتى (ا ح ط) (ح ط د) المتبادلتين متساويتين  فاقول ان خطى (ا ب) (ج د) متوازيان 
yathā abarekhāyāṃ jadarekhāyām hajharekhā saṃpātaṃ karoti | tatra ahajhakoṇo dajhahakoṇena samāno yadi jātas  tadā abarekhā jadarekhā ca samānāntarā bhavati | 
Εἰ γὰρ μή, ἐκβαλλόμεναι αἱ ΑΒ, ΓΔ συμπεσοῦνται ἤτοι ἐπὶ τὰ Β, Δ μέρη ἢ ἐπὶ τὰ Α, Γ.  ἐκβεβλήσθωσαν καὶ συμπιπτέτωσαν ἐπὶ τὰ Β, Δ μέρη κατὰ τὸ Η.  τριγώνου δὴ τοῦ ΗΕΖ ἡ ἐκτὸς γωνία ἡ ὑπὸ ΑΕΖ ἴση ἐστὶ τῇ ἐντὸς καὶ ἀπεναντίον τῇ ὑπὸ ΕΖΗ: ὅπερ ἐστὶν ἀδύνατον:  οὐκ ἄρα αἱ ΑΒ, ΓΔ ἐκβαλλόμεναι συμπεσοῦνται ἐπὶ τὰ Β, Δ μέρη.  ὁμοίως δὴ δειχθήσεται, ὅτι οὐδὲ ἐπὶ τὰ Α, Γ:  αἱ δὲ ἐπὶ μηδέτερα τὰ μέρη συμπίπτουσαι παράλληλοί εἰσιν:  παράλληλος ἄρα ἐστὶν ἡ ΑΒ τῇ ΓΔ. 
For, if not, AB, CD when produced will meet either in the direction of B, D or towards A, C.  Let them be produced and meet, in the direction of B, D, at G.  Then, in the triangle GEF, the exterior angle AEF is equal to the interior and opposite angle EFG: which is impossible. [I. 16]  Therefore AB, CD when produced will not meet in the direction of B, D.  Similarly it can be proved that neither will they meet towards A, C.  But straight lines which do not meet in either direction are parallel; [Def. 23]  therefore AB is parallel to CD. 
Si enim non, educte recte AB et GD concident vel in partes BD vel in AG.  Educantur et concidant in BD ad punctum I.  Trigoni ergo IEZ exterior angulus AEZ equalis est interiori et opposito EZI. Quod est impossibile.  Non ergo recte AB et GD educte concidunt in partes BD.  Similiter autem ostendetur quoniam neque in partes AG.  In neutras vero partes concidentes equidistantes sunt.  Equidistans ergo est recta AB recte GD. 
برهانه انهما ان لم يكونا متوازيين فانهما اذا اخرجا فى احدى الجهتين التقيا  فنخرجهما فى جهة (ب د) فيلتقيان على نقطة (ك) ان امكن ذلك  فتصير ١١٦ زاوية (ا ح ط) الخارجة من مثلث (ح ط ك) اعظم من زاوية (ح ط ك) الداخلة كما بين ببرهان (يو) من (ا) وهذا خلف لان زاوية (ا ح ط) فرضت مساوية لزاوية (ح ط د)  فخط (ا ب) (ج د) ان اخرجا فى الجهتين جميعا لم يلتقيا ولو خرج الى غير نهاية    فهما متوازيان  ... 
yadi ca rekhe samānāntare na bhaviṣyataḥ  tadā ubhe rekhe vārddhite vacinhe miliṣyati |  tatra vahajhatribhujaṃ bhaviṣyati | evaṃ tribhujād bahisthaḥ ahajhakoṇas tribhujāntargataḥ hajhavakoṇaś caitau tulyau syātām | idam anupapannam |        tasmād rekhādvayaṃ samānāntarakaṃ bhavatīti siddham || 
Ἐὰν ἄρα εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐναλλὰξ γωνίας ἴσας ἀλλήλαις ποιῇ, παράλληλοι ἔσονται αἱ εὐθεῖαι:  ὅπερ ἔδει δεῖξαι. 
Therefore etc.  Q. E. D. 
Si ergo in duas rectas recta incidens etc.  Quod oportebat (ostendere). 
...  وذلك ما اردنا ان نبين. 
   
 
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