Bibliotheca Polyglotta

In any triangle the greater side subtends the greater angle. For let ABC be a triangle having the side AC greater than AB; I say that the angle ABC is also greater than the angle BCA.

OMNIS trianguli maius latus maiorem angulum subtendit. IN triangulo A B C, sit latus A C, maius latere A B. Dico angulum A B C, subtensum a maiori latere A C, maiorem esse angulo A C B, qui a minori latere A B, subtenditur.

凡三角形。大邊對大角。小邊對小角。 . .

For, since AC is greater than AB, let AD be made equal to AB [I. 3], and let BD be joined.

Nam ex A C, auferatur A D, æqualis ipsi A B, & ducatur recta B D.

Then, since the angle ADB is an exterior angle of the triangle BCD, it is greater than the interior and opposite angle DCB. [I. 16] But the angle ADB is equal to the angle ABD, since the side AB is equal to AD; therefore the angle ABD is also greater than the angle ACB; therefore the angle ABC is much greater than the angle ACB.

Quonim igitur duo latera A B, A D, æqualia sunt per constructionem, erunt anguli A B D, A D B, æquales: Est autem angulus A D B, maior angulo A C B. . Igitur & angulus A B D, maior erit angulo A C B.¹ Quamobrem cum angulus totus A B C, maior adhuc sit angulo A B D; erit angulus A B C, multo maior angulo A C B. Eadem ratione, si latus A C, maius ponatur latere B C, ostendens angulum A B C, maiorem esse angulo B A C; si nimirum ex C A, abscindatur linea æqualis ipsi C B, &c.

1. Clavius applies the same arguments, arranged in the same order, as Gerardius.

. . . .

Therefore etc. Q. E. D.

Quare omnis trianguli maius latus maiorem angulum subtendit; Quod demonstrandum erat.

COROLLARIVM
EX hoc sequitur, omnes tres angulos trianguli Scaleni esse inæquales, ut monuimus defin. 15. huius lib. Sit enim triangulum Scalenum A B C, cuius maximum quidem latus A C, minimum autem B C, & medium locum habens A B. Dico eiusdem omnes angulos inæquales esse. Cum enim latus A C, ponatur maius latere A B, erit per hanc propositio angulus B, angulo C, maior. Eadem ratione maior erit angulus C, angulo A, quandoquidem & latus A B, latere B C, maius ponitur. Sunt igitur omnes tres anguli inæquales, maximus quidem B, minimus vero A, & C, medium locum inter utrumque tenens.

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