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Euclid: Elementa
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Proposition 30. 
THEOR. 21. PROPOS. 30. 
第三十題 
Straight lines parallel to the same straight line are also parallel to one another. 
QVÆ eidem rectæ lineæ parallelæ, & inter se sunt parallelæ. 
兩直線。與他直線平行。則元兩線亦平行。 
Let each of the straight lines AB, CD be parallel to EF;  I say that AB is also parallel to CD. 
SINT rectæ A B, C D, eidem rectæ E F, parallelæ.  Dico & ipsas A B, C D, esse inter se parallelas. 
.  . 
For let the straight line GK fall upon them; 
Quoniam enim omnes hæ lineæ in eodem ponuntur esse plano. (Nam propositio 9. undecimi libri agetur de lineis in diuersis planis) ducta recta G H, secabis omnes, nimirum A B, in I; C D, in K; & E F, in L. 
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Then, since the straight line GK has fallen on the parallel straight lines AB, EF, the angle AGK is equal to the angle GHF. [I. 29]  Again, since the straight line GK has fallen on the parallel straight lines EF, CD,the angle GHF is equal to the angle GKD. [I. 29]  But the angle AGK was also proved equal to the angle GHF;  therefore the angle AGK is also equal to the angle GKD; [C.N. 1]  and they are alternate.  Therefore AB is parallel to CD. 
Quia igitur A B, ponitur parallela ipsi E F, erit angulus A I L, alterno F L I, æqualis.  Rursus quia C D, ponitur etiam parallela ipsi E F, erit angulus D K I, eidem angulo F L I, nempe internus externo, vel externus interno, æqualis.    Quare anguli A I L, D K I, æquales inter se quoque erunt.  Cum igitur sint alterni,  erunt rectæ A B, C D, parallelæ inter se. 
.  .  .  .  .  . 
  Q. E. D. 
Quæ igitur eidem rectæ lineæ parallelæ, & inter se sunt parallelæ.  Quod demonstrandum erat.

SCHOLION
QVOD si quis dicat, duas rectas A I, B I, parallelas esse rectæ C D, & tamen ipsas non esse parallelas; Occurrendum est, duas A I, B I, non esse duas lineas, sed partes tantum unius lineæ. Concipiendum enim est animo, quaslibet parallelas infinite esse productas; Constat autem A I, productam coincidere cum B I. Quamobrem quæ eidem rectæ lineæ parallelæ, & inter se sunt parallelæ: vel certe, quando inter se coeunt. Quod ita demonstrabitur. Sint duæ rectæ A B, A C, coeuntes in A, parallelæ ipsi D E. Dico illas in rectum esse constitutas. Ex puncto enim A, ducatur recta A F, secans D E, in F, utcunque. Quoniam igitur A B, D E, sunt parallelæ, erunt anguli alterni B A F, A F E, æquales. Addito ergo communi angulo C A F, erunt duo anguli ad A, æquales duobus angulis C A F, A F E. Sed hi duo æquales sunt duobus rectis, cum sint interni inter duas parallelas A C, D E. Igitur & duo anguli ad A, duobus eruntrectis æquales; ac propterea in rectum erunt constitutæ ipsæ AB, AC. Quod est propositum. 
.  . 
 
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