You are here: BP HOME > BPG > Euclid: Elementa > fulltext
Euclid: Elementa

Choose languages

Choose images, etc.

Choose languages
Choose display
  • Enable images
  • Enable footnotes
    • Show all footnotes
    • Minimize footnotes
Search-help
Choose specific texts..
    Click to Expand/Collapse Option Complete text
Click to Expand/Collapse OptionTitle
Click to Expand/Collapse OptionPreface
Click to Expand/Collapse OptionBook I
Click to Expand/Collapse OptionBook ΙI
Click to Expand/Collapse OptionBook IΙΙ
Click to Expand/Collapse OptionBook IV
Click to Expand/Collapse OptionBook V
Click to Expand/Collapse OptionBook VI
Click to Expand/Collapse OptionBook VII
Click to Expand/Collapse OptionBook VIII
Click to Expand/Collapse OptionBook ΙΧ
Click to Expand/Collapse OptionBook Χ
Click to Expand/Collapse OptionBook ΧI
Click to Expand/Collapse OptionBook ΧIΙ
Click to Expand/Collapse OptionBook ΧIΙΙ
β῾ 
PROPOSITION 2. 
 
 
第二題 二支 
Ἐὰν τριγώνου παρὰ μίαν τῶν πλευρῶν ἀχθῇ τις εὐθεῖα, ἀνάλογον τεμεῖ τὰς τοῦ τριγώνου πλευράς: καὶ ἐὰν αἱ τοῦ τριγώνου πλευραὶ ἀνάλογον τμηθῶσιν, ἡ ἐπὶ τὰς τομὰς ἐπιζευγνυμένη εὐθεῖα παρὰ τὴν λοιπὴν ἔσται τοῦ τριγώνου πλευράν. 
If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle. 
Si qua secundum unum laterum trigoni recta trahatur, proportionaliter secat trigoni latera. Et si trigoni latera proportionaliter divisa fuerint, in sectiones copulata recta secundum reliquum erit trigoni latus. 
 
 
三角形。任依一邊作平行線。卽此線分兩餘邊以為比例。必等。三角形內。有一線分兩邊以為比例、而等。卽此線與餘邊為平行。 
Τριγώνου γὰρ τοῦ ΑΒΓ παράλληλος μιᾷ τῶν πλευρῶν τῇ ΒΓ ἤχθω ἡ ΔΕ:  λέγω, ὅτι ἐστὶν ὡς ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ. 
For let DE be drawn parallel to BC, one of the sides of the triangle ABC;  I say that, as BD is to DA, so is CE to EA. 
Trigoni enim ABG parallilos uni laterum quod sit BG trahatur recta DE.  Dico quoniam est ut BD ad DA ita GE ad EA. 
   
   
   
Ἐπεζεύχθωσαν γὰρ αἱ ΒΕ, ΓΔ. 
For let BE, CD be joined. 
Copulentur enim recte BE, GD. 
 
 
 
Ἴσον ἄρα ἐστὶ ΒΔΕ τρίγωνον τῷ ΓΔΕ τριγώνῳ:  ἐπὶ γὰρ τῆς αὐτῆς βάσεώς ἐστι τῆς ΔΕ καὶ ἐν ταῖς αὐταῖς παραλλήλοις ταῖς ΔΕ, ΒΓ:  ἄλλο δέ τι τὸ ΑΔΕ τρίγωνον.  τὰ δὲ ἴσα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον:  ἔστιν ἄρα ὡς τὸ ΒΔΕ τρίγωνον πρὸς τὸ ΑΔΕ [τρίγωνον], οὕτως τὸ ΓΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον.  ἀλλ᾽ ὡς μὲν τὸ ΒΔΕ τρίγωνον πρὸς τὸ ΑΔΕ, οὕτως ἡ ΒΔ πρὸς τὴν ΔΑ:  ὑπὸ γὰρ τὸ αὐτὸ ὕψος ὄντα τὴν ἀπὸ τοῦ Ε ἐπὶ τὴν ΑΒ κάθετον ἀγομένην πρὸς ἄλληλά εἰσιν ὡς αἱ βάσεις.  διὰ τὰ αὐτὰ δὴ ὡς τὸ ΓΔΕ τρίγωνον πρὸς τὸ ΑΔΕ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ:  καὶ ὡς ἄρα ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ. 
Therefore the triangle BDE is equal to the triangle CDE;  for they are on the same base DE and in the same parallels DE, BC. [I. 38]  And the triangle ADE is another area.  But equals have the same ratio to the same; [V. 7]  therefore, as the triangle BDE is to the triangle ADE, so is the triangle CDE to the triangle ADE.  But, as the triangle BDE is to ADE, so is BD to DA;  for, being under the same height, the perpendicular drawn from E to AB, they are to one another as their bases. [VI. 1]  For the same reason also, as the triangle CDE is to ADE, so is CE to EA.  Therefore also, as BD is to DA, so is CE to EA. [V. 11] 
Equale ergo est trigonum BDE trigono GDE.  Super eandem enim basim sunt que est DE et in eisdem equidistantibus que sunt DE, BG.  Aliud autem quoddam trigonum ADE.  Equalia vero ad idem eandem habent proportionem.  Est ergo ut BDE trigonum ad ADE trigonum ita GDE ad ADE trigonum.  Verum ut trigonum quidem BDE ad ADE ita recta BD ad DA,  sub eadem enim altitudine existentia ab E in AB ducta catheto ad se invicem sunt ut bases.  Propter eadem ergo ut GDE trigonum ad ADE ita recta GE ad EA.  Et sicut ergo recta BD ad DA ita recta GE ad EA. 
                 
                 
                 
Ἀλλὰ δὴ αἱ τοῦ ΑΒΓ τριγώνου πλευραὶ αἱ ΑΒ, ΑΓ ἀνάλογον τετμήσθωσαν,  ὡς ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ,  καὶ ἐπεζεύχθω ἡ ΔΕ:  λέγω, ὅτι παράλληλός ἐστιν ἡ ΔΕ τῇ ΒΓ. 
Again, let the sides AB, AC of the triangle ABC be cut proportionally,  so that, as BD is to DA, so is CE to EA;  and let DE be joined.  I say that DE is parallel to BC. 
Sed et trigoni ABG latera AB, AG proportionaliter secta sunt secundum D, E puncta.  Ut ergo BD ad DA ita GE ad EA.  Et copuletur recta DE.  Dico quoniam parallilos est recta DE recte BG. 
       
       
       
Τῶν γὰρ αὐτῶν κατασκευασθέντων,  ἐπεί ἐστιν ὡς ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ,  ἀλλ᾽ ὡς μὲν ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως τὸ ΒΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον,  ὡς δὲ ἡ ΓΕ πρὸς τὴν ΕΑ, οὕτως τὸ ΓΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον,  καὶ ὡς ἄρα τὸ ΒΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον, οὕτως τὸ ΓΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον.  ἑκάτερον ἄρα τῶν ΒΔΕ, ΓΔΕ τριγώνων πρὸς τὸ ΑΔΕ τὸν αὐτὸν ἔχει λόγον.  ἴσον ἄρα ἐστὶ τὸ ΒΔΕ τρίγωνον τῷ ΓΔΕ τριγώνῳ:  καί εἰσιν ἐπὶ τῆς αὐτῆς βάσεως τῆς ΔΕ.  τὰ δὲ ἴσα τρίγωνα καὶ ἐπὶ τῆς αὐτῆς βάσεως ὄντα καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν.  παράλληλος ἄρα ἐστὶν ἡ ΔΕ τῇ ΒΓ. 
For, with the same construction,  since, as BD is to DA, so is CE to EA,  but, as BD is to DA, so is the triangle BDE to the triangle ADE,  and, as CE is to EA, so is the triangle CDE to the triangle ADE, [VI. 1]  therefore also, as the triangle BDE is to the triangle ADE, so is the triangle CDE to the triangle ADE. [V. 11]  Therefore each of the triangles BDE, CDE has the same ratio to ADE.  Therefore the triangle BDE is equal to the triangle CDE; [V. 9]  and they are on the same base DE.  But equal triangles which are on the same base are also in the same parallels. [I. 39]  Therefore DE is parallel to BC. 
Eisdem enim dispositis  quoniam est ut recta BD ad DA ita recta GE ad EA.  verum ut BD quidem ad DA ita BDE trigonum ad ADE,  ut autem GE ad EA ita GDE ad ADE trigonum,  et sicut ergo BDE trigonum ad ADE trigonum ita GDE trigonum ad ADE trigonum.  Utrumque ergo trigonorum BDE, GDE ad ADE eandem habet proportionem.  Equale ergo est BDE trigonum trigone GDE  et sunt super eandem basim DE.  Equalia vero trigona super eandem basim existentia, et in eisdem equidistantibus sunt.  Parallilos ergo est recta DE recte BG. 
                   
                   
                   
Ἐὰν ἄρα τριγώνου παρὰ μίαν τῶν πλευρῶν ἀχθῇ τις εὐθεῖα, ἀνάλογον τεμεῖ τὰς τοῦ τριγώνου πλευράς: καὶ ἐὰν αἱ τοῦ τριγώνου πλευραὶ ἀνάλογον τμηθῶσιν, ἡ ἐπὶ τὰς τομὰς ἐπιζευγνυμένη εὐθεῖα παρὰ τὴν λοιπὴν ἔσται τοῦ τριγώνου πλευράν:  ὅπερ ἔδει δεῖξαι. 
Therefore etc.  Q. E. D. 
Si qua ergo secundum unum laterum etc.  Quod oportet ostendere. 
   
   
   
 
Go to Wiki Documentation
Enhet: Det humanistiske fakultet   Utviklet av: IT-seksjonen ved HF
Login