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..
Euclid: Elementa
    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
Back to library
Proposition 38. 
THEOR. 28. PROPOS. 38. 
第三十八題 
Triangles which are on equal bases and in the same parallels are equal to one another. 
TRIANGVLA super æqualibus basibus constituta, & in eisdem parallelis, inter se sunt æqualia. 
兩平行線內。有兩三角形。若底等。則兩形必等。 
Let ABC, DEF be triangles on equal bases BC, EF and in the same parallels BF, AD;  I say that the triangle ABC is equal to the triangle DEF. 
INTER parallelas A B, C D, & super æquales bases C E, F D, sint constituta triangula A C E, B F D.  Dico ipsa esse æqualia. 
.  . 
For let AD be produced in both directions to G, H;  through B let BG be drawn parallel to CA, [I. 31] and through F let FH be drawn parallel to DE.  Then each of the figures GBCA, DEFH is a parallelogram; and GBCA is equal to DEFH;  for they are on equal bases BC, EF and in the same parallels BF, GH. [I. 36]  Moreover the triangle ABC is half of the parallelogram GBCA; for the diameter AB bisects it. [I. 34]  And the triangle FED is half of the parallelogram DEFH; for the diameter DF bisects it. [I. 34]  [But the halves of equal things are equal to one another.]  Therefore the triangle ABC is equal to the triangle DEF. 
  Ducatur enim E G, parallela ipsi A C, & D H, ipsi B F.  Eruntque parallelogramma A C E G, B F D H, æqualia.    Cum igitur horum dimidia sint triangula A C E, B E D;      erunt hæc inter se æqualia. 
.  .  .  .  .  .  .  . 
Therefore etc.  Q. E. D. 
Triangula ergo super æqualibus basibus, &c.  Quod erat ostendendum.

SCHOLION
CONVERSA huius ostendetur ab Euclide propos. 40.
COLLIGITVR autem ex hac propositione, si a quouis angulo trianguli dati linea recta ducatur diuidens latus oppositum bifariam, triangulum quoque bifariam secari. Ducatur enim in triangulo A B C, ex angulo A, recta A D, diuidens bifariam latus B C, in D. Dico triangulum A B C, bifariam quoque secari. Si enim per A, ducatur parallela ipsi B C, erunt duo triangula A B D, A D C, inter easdem parallelas; & super æquales bases. Quare æqualia erunt.

EX PELETARIO
A puncto quouis dato in uno latere trianguli propositi lineam rectam ducere, quæ bifariam secet triangulum datum.
SIT triangulum A B C, & punctum datum D, in latere B C. Oportet igitur ex D, rectam lineam ducere, quæ bifariam diuidat triangulum. Quod si punctum D, diuidat latus B C, bifartam, recta D A, ducta ad A, diuidet triangulum bifariam, ut in hoc scholio est ostensum: Si vero D, non diuidit B C, bifariam, secetur B C, bifariam in E. Deinde ex D, ad angulum oppositum A ducatur recta D A, & per E, parallela E F, ipsi D A, secans A C, in F. Si igitur ducatur recta D F, erit triangulum diuisum bifariam a linea D F. Nam ducta recta E A, erunt triangula E F A, E F D, æqualia, cum sint super eandem basin E F, & inter easdem parallelas E F, A D. Addito igitur communi C F E, erunt tota triangula A E C, C D F, æqualia: Est autem A E C, dimidium totius A B C, ut iam fuit ostensum. Igitur & C D F, dimidium est eiusdem trianguli A B C. quod erat probandum.
QVOD si punctum D, fuerit in altera medietate E C, eodem modo problema conficiemus: sed tunc triangulum abscindetur ad partes B, trapezium vero ad partes C, ut figura præsens satis indicat. Demonstratio autem eadem est, si in ea mutetur litera B, in C, & C, in B. Hoc tamen problema multo nos uniuersalius proponemus ad finem sexti libri. 
.  . 
 
Permanent link
Go to Wiki Documentation
Enhet: Det humanistiske fakultet   Utviklet av: IT-seksjonen ved HF
Login