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Euclid: Elementa
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Proposition 21. 
THEOR. 14. PROPOS. 21. 
第二十一題 
If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle. 
SI super trianguli uno latere, ab extremitatibus duæ rectæ lineæ interius constitutæ fuerint; hæ constitutæ reliquis trianguli duobus lateribus minores quidem erunt, maiorem vero angulum continebunt. 
凡三角形。於一邊之兩界、出兩線。復作一三角形、在其內。則內形兩腰。幷之必小於相對兩腰。而後兩線所作角。必大於相對角。 
On BC, one of the sides of the triangle ABC, from its extremities B, C, let the two straight lines BD, DC be constructed meeting within the triangle;  I say that BD, DC are less than the remaining two sides of the triangle BA, AC, but contain an angle BDC greater than the angle BAC. 
IN triangulo A B C, super extremitates B, & C, lateris B C, intra triangulum constituantur duæ rectæ lineæ B D, C D, in puncto D, concurrentes.  Dico B D, C D, simul minores esse duobus lateribus B A, C A, simul; At vero angulum B D C, maiorem angulo B A C. 
.  . 
For let BD be drawn through to E.  Then, since in any triangle two sides are greater than the remaining one, [I. 20]  therefore, in the triangle ABE, the two sides AB, AE are greater than BE.  Let EC be added to each;  therefore BA, AC are greater than BE, EC.  Again, since, in the triangle CED, the two sides CE, ED are greater than CD, let DB be added to each;  therefore CE, EB are greater than CD, DB.  But BA, AC were proved greater than BE, EC;  therefore BA, AC are much greater than BD, DC. 
Producatur enim altera linearum interiorum, nempe B D, ad punctum E, lateris C A.    Quoniam igitur in triangulo B A E, duo latera B A, A F, maiora sunt latere B E,  si addatur commune E C,  erunt B A, A C, maiora quam B E, E C.  Rursus quia in triangulo C E D, duo latera C E, E D, maiora sunt latere C D; si commune apponatur D B,  erunt C E, E B, maiora quam C D, D B.  Ostensum vero iam fuit, A B, C A, maiora esse quam B E, E C.  Multo igitur maiora erunt B A, C A, quam B D, C D, quod primo proponebatur. 
.  .  .  .  .  .  .  .  . 
Again, since in any triangle the exterior angle is greater than the interior and opposite angle, [I. 16]  therefore, in the triangle CDE, the exterior angle BDC is greater than the angle CED.  For the same reason, moreover, in the triangle ABE also, the exterior angle CEB is greater than the angle BAC.  But the angle BDC was proved greater than the angle CEB;  therefore the angle BDC is much greater than the angle BAC. 
  Præterea, quoniam angulus B D C, maior est angulo D E C, externus interno;  & angulus D E C, angulo B A C, maior quoque est, eandem ob causam:    Erit angulus B D C, multo maior angulo B A C; quod secundo proponebatur. 
.  .  .  .  . 
Therefore etc.  Q. E. D. 
Si igitur super trianguli uno latere, ab extremitatibus, &c.  Quod erat ostendendum.

SCHOLION
QVAM recte Euclides dixerit, duas illas lineas intra triangulum constitutas, duci debere ab extremitatibus unius lateris, aperte intelligi potest ex eo, quod mox ex Proclo demonstrabimus; in triangulis videlicet rectangulis, vel etiam amblygoniis, intra triangulum constitui posse duas lineas super unum latus circa angulum rectum, vel obtusum, quarum quidem una ab extremitate dicti lateris, altera vero a quouis puncto prope aliud extremum lateris eiusdem educitur, quæ maiores sint reliquis duobus trianguli lateribus. Item in triangulis scalenis eodem modo super maximum latus duas rectas intra triangulum constitui posse, quæ minorem comprehendant angulum, &c.

EX PROCLO
SIT triangulum habens exempli gratia angulum A B C, obtusum. Dico ab extremo C, & a quouis puncto, nempe a D, prope aliud extremum B, lateris B C, duci posse duas lineas intra triangulum ad aliquod punctum, quæ maiores sint duobus lateribus B A, A C. Ducatur enim recta D A: Et quoniam in triangulo A B D, duo anguli A B D, A D B, minores sunt duobus rectis. Ponitur autem A B D, maior recto, nempe obtusus; erit A D B minor recto, ideoque minor angulo A B D. Quare latus A D, maius erit latere A B. Ex D A, abscindatur recta D E, æqualis rectæ A B. Et reliqua linea A E, bifariam diuidatur in F. Si igitur ab extremo C, ad F, recta dueatur C F, erunt duæ lineæ rectæ constitutæ C F, D F, intra triangulum maiores duobus lateribus B A, A C. Quoniam enim in triangulo A F C, duo latera A F, F C, maiora sunt latere A C. Est autem recta A F, ipsi F E, æqualis, per constructionem; erunt C F, F E, maiores quoque latere C A. Si igitur æqualia addantur E D, & A B, fient rectæ C F, F D, maiores lateribus C A, A B. Quod est propositum. Quod si ad F, ex B, extremo recta duceretur, essent duæ rectæ constitutæ C F, B F, minores duobus lateribus C A, A B, ut Euclides demontrauit.
RVRSVS sit triangulum scalenum A B C, cuius latus maximum B C, minimum A B. Ex B C, auferatur B D, æqualis rectæ A B, & ducatur A D, recta, ad cuius punctum quodlibet, ut ad E, ab extremo C, recta ducatur C E. Constitutæ igitur erunt intra triangulum duæ lineæ C E, D E, quæ minorem angulum comprebendunt eo, quem efficiunt duo latera A B, A C. Cum enim duo tatera B A, B D, æqua lia sint, erunt duo anguli B A D, B D A, æquales: Sed B D A, angulus maior est angulo C E D. Maior igitur erit & angulus B A D, angulo C E D. Quare multo maior erit totus angulus B A C, angulo C E D. Quod est propositum. Recte igitur Euclides monuit, duas lineas intra triangulum constitutas educi debere, ab extremis punctis unius lateris, ut minores quidem sint duob us reliquis trianguli lateribus, maiorem vero complectantur angulum. Alias enim propositio vera non esset, ut iam est demonstratum. 
.  . 
 
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