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Euclid: Elementa
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Proposition 8. 
THEOR. 5. PROPOS. 8. 
第八題 
If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines. 
SI duo triangula duo latera habuerint duobus lateribus; vtrumque vtrique æqualia, habuerint verò & basim basi æqualem: Angulum quoque sub æqualibus rectis lineis contentum angulo æqualem habebunt. 
兩三角形。若相當之兩腰各等。兩底亦等。則兩腰間角必等。 
Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF;  and let them have the base BC equal to the base EF;  I say that the angle BAC is also equal to the angle EDF. 
SINT duo latera A B, A C, trianguli A B C, duobus lateribus D E, D F, trianguli D E F, æqualia, vtrumque vtrique, nempe A B, ipsi D E, & A C, ipsi D F;   sit autem & basis B C, basi E F, æqualis.   Dico angulum A, æqualem esse angulo D, quorum videlicet vterque dictis lateribus continetur.  
.  .  . 
For, if the triangle ABC be applied to the triangle DEF, and if the point B be placed on the point E and the straight line BC on EF, the point C will also coincide with F, because BC is equal to EF.  Then, BC coinciding with EF, BA, AC will also coincide with ED, DF;  for, if the base BC coincides with the base EF, and the sides BA, AC do not coincide with ED, DF but fall beside them as EG, GF,  then, given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there will have been constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it.  But they cannot be so constructed. [I. 7]  Therefore it is not possible that, if the base BC be applied to the base EF, the sides BA, AC should not coincide with ED, DF;  they will therefore coincide, so that the angle BAC will also coincide with the angle EDF, and will be equal to it. 
Nam si mente intelligatur basis B C, superponi basi E F, neutra excedet alteram, sed punctum B, congruet puncto E, & punctum C, puncto F, cum hæ bases ponantur æquales inter se.     Deinde si triangulum A B C, cogitetur cadere super triangulum D E F, cadet punctum A, aut in ipsum punctum D, aut alio. Si punctum A, in ipsum punctum D, cadat, congruent sibi mutuo triangulorum latera, cum ponantur æqualia; Ac propterea angulus A, æqualis erit angulo D, cum neuter alterum excedat.  Quod si punctum A, alio dicatur cadere, ut ad G, quomodocunque id contingat, hoc est, sive in latus E D, sive intra triangulum E D F, sive extra, ut in figuris apparet; erit perpetuo E G, (quæ eadem est, quæ B A,) æqualis ipsi E D; & F G, (quæ eadem est, quæ C A) æqualis ipsi F D, propterea quod latera unius trianguli æqualia ponuntur lateribus alterius.    Hoc autem fieri non posse, iamdudum demonstratum est, cum tam rectæ E G, E D, terminum eundem E, quam rectæ F G, F D, eundem limitem F, possideant.   Non igitur punctum A, cadet alio quam in punctum D: ac propterea angulus A, angulo D, æqualis erit. 
.  .  .  .  .  .  . 
If therefore etc.  Q. E. D. 
Quare si duo triangula duo latera habuerint duobus lateribus, &c.   Quod erat demonstrandum.

SCHOLION
VT vides, hæc propositio conuertit primam partem propositionis quartæ. Sicut enim ibi ex æqualitate angulorum, qui lateribus æqualibus continentur, collecta fuit basium æqualitas; ita hic ex æqualitate basium concludit Euclides æqualitatem angulorum, qui lateribus æqualibus comprehenduntur. Possumus eodem modo ex prima, & tertia parte conclusionis quartæ propositionis inferre totum antecedens eiusdem, ita ut theorema proponatur in hanc formam.
SI duo triangula bases habuerint æquales, & angulos super bases constitutos æquales, utrumque utrique: Habebunt quoque reliqua latera æqualia, utrumque utrique, quæ videlicet æqualibus angulis subtenduntur, angulosque reliquos hisce lateribus inclusos æquales.
SIT enim basis B C, æqualis basi E F, & angulus B, angulo E, angulusque C, angulo D F E. Dico latus quoque A B, lateri D E, & latus A C, lateri D F, æquale esse, angulumque A, angulo D. Nam si basis basi superponatur, congruent sibi mutuo extrema earum, nec non & lineæ angulorum æqualium. Quare omnia sibi congruent, proptereaque omnia inter sese æqualia erunt. Verum hoc idem theorema a nobis propositum, quod quidem magis proprie conuertere videtur quartam propositionem, quam illud Euclidis, aliter demonstrabit Euclides in prima parte propositionis 26. ut eo loco monebimus.

COROLLARIUM
PORRO ex antecedente huius octauæ propositionis non solum colligi potest, angulos lateribus æqualibus contentos æquales esse, verum etiam reliquos angulos, qui ad bases constituuntur, utrumque utrique, ut angulum B, angulo E, & angulum C, angulo F; imo totum triangulum toti triangulo, ut constat ex eadem superpositione unius trianguli super alterum. Nam sibi mutuo congruent & dicti anguli, & tota triangula, ut perspicuum est. Quod etiam ex quarta propositio colligi poterit, postquam demonstratum fuerit, angulos æqualibus comprehensos lateribus æquales esse. Inde enim fiet, cum latera quoque sint æqualia, & reliquos angulos, & tota triangula esse æqualia, ut in propositio 4. demonstratum est.

EX PROCLO
PHILONIS familiares conantur hoc idem theorem a octavum ostenduns demonstratione affirmativa, hac ratione. Posito enim eodem antecedente, superponi intelligatur basis B C, basi E F, ita ut triangulum A B C, cadat in diuersas partes, & non super triangulum D E F, quare est triangulum A E F. Aut igitur duo latera, nempe D F, F A, constituunt unam lineam rectam, quod quidem continget, si duo anguli C, & F, recti extiterint; aut non. Si constituant unam lineam rectam, veluti D A, ita propositum concludetur. Quoniam in triangulo A E D, duo latera A E, D E, ponuntur æqualia (est enim nunc A E, recta eadem, quæ A B, quæ per hypothesin rectæ D E, æqualis est) erunt anguli A, & D, super basin A D, æquales, quod erat ostendendum. Si vero neque D E, F A, neque D E, E A, lineam rectam conficiant, ducatur ex D, ad A, linea recta D A, quæ vel cadet intra triangula, vel extra. Cadat primum intra, quod quidem accidet, quando anguli ad E, & F, sunt acuti. Quoniam igitur in triangulo A E D, duo latera A E, D E, æqualia ponuntur, erunt duo anguli E A D, E D A, æquales ad basim D A. Eadem ratione, cum duo latera A F, D F, æqualia sint per hypothesin, erunt duo anguli F A D, F D A, super basin D A, æquales. Si igitur hi æquales illis æqualibus addantur, fient toti anguli E A F, E D F, æquales. Quod erat ostendendum. Cadat deinde recta D A, extra triangula, quod demum fiet, quando anguli ad F, fuerint obtusi. Quoniam igitur in triangulo A E D, duo latera A E, D E, ponuntur æqualia, erunt anguli E A D, E D A, æquales super basin D A. Eadem ratione, cum duo latera A F, D F, in triangulo A F D, sint per hypothesin æqualia, erunt anguli F A D, F D A, super basin D A, æquales. His ergo a prioribus ablatis, remanebunt anguli E A F, E D F, æquales; Quod demonstrandum proponebatur. 
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