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Euclid: Elementa
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Proposition 7. 
THEOR. 4. PROPOS. 7. 
第七題 
Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be 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. 
SVPER eadem recta linea, duabus eisdem rectis lineis aliæ duæ rectæ lineæ æquales, vtraque vtrique, non constituentur, ad aliud atque aliud punctum, ad easdem partes, eosdemque terminos cum duabus initio ductis rectis lineis habentes. 
一線為底。出兩腰線。其相遇止有一點。不得別有腰線與元腰線等。而於此點外相遇。 
For, if possible, given two straight lines AC, CB constructed on the straight line AB and meeting at the point C, let two other straight lines AD, DB be constructed on the same straight line AB, on the same side of it, meeting in another point D and equal to the former two respectively, namely each to that which has the same extremity with it,  so that CA is equal to DA which has the same extremity A with it, and CB to DB which has the same extremity B with it; and let CD be joined. 
SVPER recta A B, constituantur ad punctum quoduis C, duæ rectæ lineæ A C, B C. Dico super eandem rectam A B, versus partem eandem C, non posse ad aliud punctum, vt ad D, constitui duas alias rectas lineas,   quæ sint æquales lineis A C, B C, vtraque vtrique, nempe A C, ipsi A D, quæ eundem habent terminum A; & B C, ipsi B D, quæ eundem etiam terminum possident B. 
.  . 
Then, since AC is equal to AD, the angle ACD is also equal to the angle ADC; [I. 5]  therefore the angle ADC is greater than the angle DCB;  therefore the angle CDB is much greater than the angle DCB.  Again, since CB is equal to DB, the angle CDB is also equal to the angle DCB.  But it was also proved much greater than it: which is impossible. 
Sint enim, si fieri potest, rectæ A C, A D, inter se, & rectæ B C, B D, inter se etiam æquales. Aut igitur punctum D, erit in alterutra rectarum A C, B C, ita vt recta A D, in ipsam rectam A C vel B D, in ipsam B C, cadat; aut intra triangulum A B C; aut extra. Sit primo punctum D, in altera rectarum A C, B C, nempe in A C, vt A D, sit pars ipsius A C. Quoniam igitur rectæ A C, A D, eundem terminum A, habentes dicuntur æquales, erit pars A D, toti A C, æqualis. Quod fieri non potest. Sit deinde punctum D, intra triangulum A B C, & ducta recta C D, producantur rectæ B C, B D, vsque ad E, & F. Quoniam igitur in triangulo A C D, ponuntur latera A C, A D, æqualia erunt anguli A C D, A D C, super basim C D, æquales;   Est autem angulus A C D, minor angulo D C E; nempe pars toto: Igitur & angulus A D C, minor erit eodem angulo D C E.   Quare angulus C D F, pars ipsius A D C, multo minor erit eodem angulo D C E.  Rursus, quia in triangulo B C D, latera B C, B D, ponuntur æqualia, erunt anguli C D F, D C E, sub basi C D, æquales.  Ostensum autem fuit, quod idem angulus C D F, multo sit minor angulo D C E. Idem ergo angulus C D F, & minor est angulo D C E, & eidem æqualis, quod est absurdum. Sit postremo punctum D, extra triangulum A B C. Aut igitur in tali erit loco, vt vna linea super alteram cadat, vt in priori figura, dummodo loco D, intelligas C, & loco C, ipsum D; ex quo rursus colligetur pars æqualis toti, quod est absurdum. Aut in tali erit loco, vt posteriores duæ lineæ ambiant priores duas, ceu in posteriori figura, si modo loco D, iterum intelligas C, & D, loco C. Quo posito, inidem absurdum incidemus, nempe angulum D C F, & minorem esse angulo C D E, & eidem æqualem, vt perspicuum est. Aut denique punctum D, ita erit extra triangulum A B C, vt altera linearum posteriorum, nempe A D, secet alteram priorum, vt ipsam B C. Ducta igitur recta C D, cum in triangulo A C D, latera A C, A D, ponantur æqualia, erunt anguli A C D, A D C, supra basim C D, æquales: Ac proinde cum angulus A D C, minor sit angulo B D C, pars toto, erit & angulus A C D, minor eodem angulo B D C. Quare multo minor erit angulus B C D, pars anguli A C D, angulo eodem B D C. Rursus, cum in triangulo B D C, latera B C, B D, ponantur æqualia, erunt anguli B C D, B D C, super basim C D, æquales: Est aut etiam ostensum, angulum B C D, multo esse minorem angulo B D C. Idem igitur angulus B C D, & minor est angulo B C D, & eidem æqualis, quod est absurdum. Non ergo æquales sunt inter se A C, A D, & inter se quoque B C, B D.  
.  .  .  .  . 
Therefore etc.  Q. E. D. 
Quare super eadem recta linea, duabus eisdem rectis lineis, &c.   Quod erat demonstrandum.

SCHOLION
FIERI potest, vt duæ lineæ A D, B D, æquales sint duabus A C, B C, vtraque vtrique, vt A D, ipsi B C, & B D, ipsi A C, vt vltima figura indicat. Verum hoc modo non egrediuntur ab eodem puncto lineæ illæ, quæ sunt æquales inter se, vt constat. Solæ enim A C, A D, eundem limitem possident A; Item B C, B D, eundem B; optimeque demonstratum fuit ab Euclide, fieri non posse, vt A C, A D, inter se sint æquales, ita vt B C, B D, quoque inter se æquales existant. Recte igitur in propositione apposita sunt hæc verba: eosdemque terminos cum duabus initio ductis rectis lineis habentes. Rursus possunt esse duæ lineæ simul sumptæ A D, B D, æquales duabus lineis A C, B C, simul sumptis, vt in eadem figura perspici potest: Sed hoc non ostendit Euclides fieri non posse. Dixit enim non posse vtramque vtrique esse æqualem, &c.
Eadem ratione possunt ex A, & B, infra A B, basim trianguli A B C, hoc est, ad contrarias partes, duci duæ lineæ rectæ A D, B D, conuententes ad aliquod punctum, ita vt A D, exiens e puncto A, æqualis sit ipsi A C; & B D, egrediens ex B, æqualis ipsi B C, vt perspicuum est in apposita figura. Non igitur sine causa adiecit Euclides: ad easdem partes. Denique esse poterunt duæ lineæ A C, A D, æquales inter se eundem terminum A, possidentes; Sed hoc posite, fieri nulla ratione poterit, vt reliquæ duæ B C, B D, terminum habentes eundem B, inter se quoque sint æquales, vt in hac figura apparet, & ab Euclide est demonstratum. Apposite igitur dictum est in propositione: duabus eisdem rectis lineis aliæ duæ rectæ lineæ æquales, vtraque vtrique, &c. Quare vt plane scopus Euclidi in hac propositione propositus intelligatur, diligenter singula verba propositionis sunt ponderanda. 
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