D Book VI. a 32. I. mon to the two triangles ABC ABD; the rema ning angle ACB is equal to the remaining angle BAD": therefore the triangle ABC is equiangular to the triangle ABD, and the fides about their equal angles are proportionals; wherefore b 4. 6. the triangles are fimilar c. In the like manner, it may be de- c mo ftrated, that the triangle ADC is equiangular and fimilar to the triangle ABC: And the triangles ABD, ADC, being each equiangular and fimilar to ABC, are equiangular and fimilar to one another. Therefore, in a right angled, &c. Q. E. D. B COR. From this it is manifeft, that the perpendicular drawn, from the right angle of a right angled triangle, to the base, is a mean proportional between the fegments of the base and alio that each of the fides is a mean proportional between the bafe, and its fegment adjacent to that fide. For in the triangles BDA, ADC, BD: DA:: DA: DC; and in the triangles ABC, DBA, BC: BA:: BA: BD b; and in the triangles ABC, ACD, BC: CA: CA: CD b. 1, def. 6. FRO PROP. IX. PROB. 'ROM a given ́ftraight line to cut off any part required, that is, a part which shall be contained in it a given number of times. Let AB be the given straight line; it is required to cut off from AB, a part which fhall be contained in it a given number of times. From the point A draw a straight line AC making any angle with AB; and in AC take any point D, and take AC M 4 fuch Book VI. fuch that it fhall contain AD, as oft as AB is to contain the part, which is to be cut off from it; join BC, and draw DE parallel to it; then AE is the part required to be cut off. 22. 6. b 18. 5. c C, 5. Because ED is parallel to one of the fides of the triangle ABC, viz. to BC, CD: DA:: BE: EA a; and by compofition b, CA: AD:: BA: AE: But CA is a multiple of AD; therefore BA is the fame multiple of AE, or contains AE the fame number of times that AC contains AD; and therefore, whatever part AD is of AC, AE is the fame of AB; wherefore, from the ftraight line AB the part required is cut off. Which was to be done. a 31. 1. T PROP. X. PROB. O divide a given ftraight line fimilarly to a given divided ftraight line, that is, into parts that fhall have the fame ratios to one another which the parts of the divided given straight line have. Let AB be the straight line given to be divided, and AC the divided line; it is required to divide AB fimilarly to AC. Let AC be divided in the points D, E; and let AB, AC be placed fo as to contain any angle, and join BC, and through the points D, E, draw a DF, EG parallel to BC; and through D draw DHK parallel to AB; therefore each of the figures § 34. 1. FH, HB, is a parallelogram; wherefore DH is equalb to FG, and HK to GB: and because A HE is parallel to KC, one of the fides of the triangle DKC, CE: ED::KH: HD: But KH BG, and HD-GF; therefore, CE: ED:: BG:GF: Again, because FD is parallel to EG, one of the fides of the triangle AGE, ED: DA :: GF FA: But it has been proved that B c 2. 6. G H E K C CE CE: ED: BG: GF; therefore the given straight line AB Book VI. is divided fimilarly to AC. Which was to be done. T PROP. XI. PROB. O find a third proportional to two given ftraight Let AB, AC be the two given straight lines, and let them be placed fo as to contain any angle; it is required to find a third proportional to AB, AC. Produce AB, AC to the points D, E; and make BD equal to AČ; B and having joined BC, through D, draw DE parallel to it a. Because BC is parallel to DE, a fide of the triangle ADE, AB: b BD:: AC: CE; but BD A D AC; therefore AB: AC :: AC: CE. a 31. I. b 2.6. Wherefore to the two given straight lines AB, AC a third proportional, CE, is found. Which was to be done. PROP. XII. PROB. O find a fourth proportional to three given Toftraight lines. Let A, B, C be the three given straight lines; it is required to find a fourth proportional to A, B, C. Take two ftraight lines DE, DF, containing any angle EDF; and upon these make DG equal to A, GE equal to B, 1 Book VI. B, and DH equal to C; and having joined GH, draw EF b 2.6. E a 31. 1. parallel a to it through the point E. And because GH is parallel to EF, one of the fides of the triangle DEF, DG : GE :: DH: HF b; but DGA, GEB, and DH C; and therefore A: B:: C: HF. Wherefore to the three given ftraight lines, A, B, C a fourth proportional HF is found. Which was to be done. a II. 2. PROP. XII. PROB. O find a mean proportional between two given ftraight lines. Let AB, BC be the two given straight lines; it is required to find a mean proportional between them. Place AB, BC in a straight line, and upon AC describe the femicircle ADC, and from the point B draw BD at Becaufe the angle ADC 31. 3. angle b, and because in the right angled triangle ADC, A DB is drawn, from the right angle, perpendicular to the bafe, DB is a mean proportional c Cor. 8: 6. between AB, BC, the fegments of the bafe c; therefore be tween tween the two given ftraight lines AB, BC, a mean propor- Book VI. tional DB is found. Which was to be done. PROP. XIV. THEOR. QUAL parallelograms which have one angle of the one equal to one angle of the other, have their fides about the equal angles reciprocally proportional And parallelograms which have one angle of the one equal to one angle of the other, and their fides about the equal angles reciprocally proportional, are equal to one another. A F E D. B Let AB, BC be equal parallelograms, which have the angles at B equal, and let the fides DB, BE be placed in the fame ftraight line; wherefore alfo FB, BG are in one straight linea the fides of the parallelograms AB, BC, about the equal angles, are reciprocally propor tional; that is, DB is to BE, as GB to BF. G Complete the parallelogram FE; and because the parallelograms AB, BC are equal, and FE is another parallelogram, AB:FE:: BC: FEb; but because the paralellograms AB, FE have the fame al- BC: FE:: GB: BF ; therefore But, let the fides about the equal angles be reciprocally proportional, viz. as DB to BE, fo GB to BF; the parallelogram AB is equal to the parallelogram BC. Because, DB: BE:: GB: BF, and DB: BE:: AB: FE, and GB BF:: BC: EF, therefore AB: FE:: BC: FEd: Wherefore a 14. I. b 7.5. C T. 6. d 11. 5. |