Bibliotheca Polyglotta

Euclid: Elementa

[Πόρισμα]
Καὶ φανερόν, ὅτι, ὅτε μὲν ἐντὸς τοῦ τριγώνου πίπτει τὸ κέντρον τοῦ κύκλου, ἡ ὑπὸ ΒΑΓ γωνία ἐν μείζονι τμήματι τοῦ ἡμικυκλίου τυγχάνουσα ἐλάττων ἐστὶν ὀρθῆς: ὅτε δὲ ἐπὶ τῆς ΒΓ εὐθείας τὸ κέντρον πίπτει, ἡ ὑπὸ ΒΑΓ γωνία ἐν ἡμικυκλίῳ τυγχάνουσα ὀρθή ἐστιν: ὅτε δὲ τὸ κέντρον τοῦ κύκλου ἐκτὸς τοῦ τριγώνου πίπτει, ἡ ὑπὸ ΒΑΓ ἐν ἐλάττονι τμήματι τοῦ ἡμικυκλίου τυγχάνουσα μείζων ἐστὶν ὀρθῆς. [ὥστε καὶ ὅταν ἐλάττων ὀρθῆς τυγχάνῃ ἡ διδομένη γωνία, ἐντὸς τοῦ τριγώνου πεσοῦνται αἱ ΔΖ, ΕΖ, ὅταν δὲ ὀρθή, ἐπὶ τῆς ΒΓ, ὅταν δὲ μείζων ὀρθῆς, ἐκτὸς τῆς ΒΓ:
ὅπερ ἔδει ποιῆσαι.]

eng

And it is manifest that, when the centre of the circle falls within the triangle, the angle BAC, being in a segment greater than the semicircle, is less than a right angle; when the centre falls on the straight line BC, the angle BAC, being in a semicircle, is right; and when the centre of the circle falls outside the triangle, the angle BAC, being in a segment less than the semicircle, is greater than a right angle. [III. 31]

lat Sic

(Porisma)
Et manifestum quoniam quando quidem intra trigonum cadit centrum circuli, angulus qui sub BAG in sectione maiori semicirculo contingens minor est recto. Quando vero in recta BG, in semicirculo contingens rectus erit. Quando autem extra trigonum AB(G) centrum cadit, angulus qui sub BAG in sectione minori semicirculo contingens maior est recto. Quare et quando minor recto continget datus angulus, intra trigonum cadent recte DZ, EZ. Quando rectus, in recta BG. Quando vero maior recto, extra rectam BG. Quod oportebat ostendere.

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san

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lat Clavius

HINC manifestum est, si centrum intra triangulum cadat,¹ omnes angulos esse acutos, quoniam omnes sunt in maiori segmento circuli: si vero sit in latere BC,² angulum BAC, ei lateri oppositum, esse rectum, quod sit in semicirculo: si denique cadat extra triangulum,³ angulum oppositum BAC, obtusum esse, cum sit in minori segmento circuli.

1. 31. tertij. 2. 31. tertij. 3. 31. tertij.

kin 幾何原本 p. 196

一系。若圜心在三角形內。卽三角形為銳角形。何者。每角在圜大分之上故。若在一邊之上。卽為直角形。若在形外。卽為鈍角形。

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