ven in pofition, in a given angle HGC, for it is equal a to the a 29. 1. given angle ADC; HG is given in pofition b; but it is given b 32. dat. alfo in magnitude, because it is equal to c AD which is given inc 34. 1. magnitude; therefore because G one of the extremities of the ftraight line GH given in pofition and magnitude is given, the other extremity H is given d; and the ftraight line EAF, which d 30. dat. is drawn through the given point H parallel to BC given in pofition, is therefore given e in pofition. PROP. XXXVIII. IF a ftraight line be drawn from a given point to two parallel straight lines given in position; the ratio of the fegments between the given point and the parallels fhall be given. Let the ftraight line EFG be drawn from the given point E to the parallels AB, CD; the ratio of EF to EG is given. From the point E draw EHK perpendicular to CD; and because from a given point E the ftraight line EK is drawn to CD which is given in pofition, in a given angle EKC; EK is E FH B e 31. dat. 34 A given in pofition a; and AB, CD are given in pofition; there a 33. dat. fore b the points H, K are given: And the point E is given; b 28. dat. wherefore c EH, EK are given in magnitude, and the ratio d ofc 29. dat, them is therefore given. But as EH to EK, fo is EF to EG, be-d caufe AB, CD are parallels; therefore the ratio of EF to EG is given. 1. dat. PROP. XXXIX. 35. 36. IF the ratio of the fegments of a ftraight line between See N. a given point in it and two parallel ftraight lines, be given; if one of the parallels be given in pofition, the other is alfo given in pofition. From From the given point A, let the ftraight line AED be drawn to the two parallel ftraight lines FG, BC, and let the ratio of the fegments AE, AD be given; if one of the parallels BC be given in pofition, the other FG is alfo given in pofition. From the point A, draw AH perpendicular to BC, and let it meet FG in K and because AH is drawn from the given point A to the ftraight line BC given in pofition, and makes a a 33. dat. given angle AHD; AH is given a in and, becaufe FG, BC are parallels, as AE to AD, fo is AK to AH; and E wherefore the ratio of AK to AH is given; but AH is given d 2. dat. in magnitude, therefore d AK is given in magnitude; and it is e 30. dat. also given in position, and the point A is given; wherefore e the point K is given. And because the ftraight line FG is drawn through the given point K parallel to BC which is given in po f 31. dat. fition, therefore f FG is given in position. 37. 38. PROP. XL. See N. IF the ratio of the fegments of a straight line into which it is cut by three parallel ftraight lines, be given; if two of the parallels are given in pofition, the third alfo is given in pofition. Let AB, CD, HK be three parallel ftraight lines, of which AB, CD are given in pofition; and let the ratio of the feg ments ments GE, GF into which the ftraight line GEF is cut by the three parallels, be given; the third parallel HK is given in pofi tion. In AB take a given point L, 'and draw LM perpendicular to CD, meeting HK in N; because LM is drawn from the gi ven point L to CD which is given in pofition and makes a gi ven angle LMD; LM is given in pofitiona; and CD is given a 33. dat. in pofition, wherefore the point M is given b; and the point Lb 28. dat. is given, LM is therefore given in magnitude; and becaufe c 29. dat. the ratio of GE to GF is given, and as GE to GF, fo is NL to Cor. 7. dat. e 2. dat. NM; the ration of NL to NM is given; and therefore d the ratio of ML to LN is given; but LM is given in magnitude, d 6. or wherefore e LN is given in magnitude; and it is alfo given in pofition, and the point L is given; wherefore f the point N is given; and becaufe the ftraight line HK is drawn through the given point N parallel to CD which is given in pofition, therefore HK is given in position 8. IF PROP. XLI. f 30. dat. g 31. dat F. a straight line meets three parallel ftraight lines See N. which are given in pofition; the fegments into which they cut it have a given ratio. Let the parallel ftraight lines AB, CD, EF given in pofition be cut by the ftraight line GHK; the ratio of GH to HK is given. In AB take a given point L, and draw LM perpendicular to CD, meeting EF in N; therefore a LM is given in pofition; and CD, EF are given C_H in pofition, wherefore the points M, Nare given: And the point L is given; A GL B a 33: dat. HM D therefore b the ftraight lines LM, MN are given in magnitude; and the ratio E K c. 1. dat. of LM to MN is therefore given c: But as LM to MN, fo is GH to HK; wherefore the ratio of GH to HK is given. PROP. XLII. IF each of the fides of a triangle be given in magnitude; the triangle is given in fpecies. Let each of the fides of the triangle ABC be given in magnitude; the triangle ABC is given in species. Make a triangle a DEF the fides of which are equal, each to each, to the given ftraight lines AB, BC, CA; which can be done, becaufe any two of them must be greater than the third; and let DE be e 44 qual to AB, EF to BC, CE EDF is equal b to the angle BAC; therefore, because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is given, in like manner the angles at B, C are given. And because the fides AB, BC, CA are given, their ratios to one another are given d, therefore the triangle ABC is given e in fpecies. PROP. XLIII. IF each of the angles of a triangle be given in magnitude; the triangle is given in fpecies. Let each of the angles of the triangle ABC be given in mag、 nitude; the triangle ABC is given in fpecies. Take a ftraight line DE given in pofition and magnitude, and at the equal to the angle BAC, and the angle DEF equal to ABC; there, fore the other angles EFD, BCA B A D CEF are equal; and each of the angles at the points A, B, C, is gi ven, wherefore each of those at the points D, E, F is given: And because the straight line FD is drawn to the given point Din DE which is given in pofition, making the given angle EDF; therefore DF is given in pofition b. In like manner EFb 32. dat. alfo is given in pofition; wherefore the point F is given: And the points D, E are given; therefore each of the ftraight lines DE, EF, FD is given e in magnitude; wherefore the triangle c 29. dat. DEF is given in fpecies d; and it is fimilar to the triangled 42. dat. ABC; which therefore is given in species. IF PROP. XLIV. one of the angles of a triangle be given, and if the fides about it have a given ratio to one another; the triangle is given in fpecies. Let the triangle ABC have one of its angles BAC given, and let the fides BA, AC about it have a given ratio to one another; the triangle ABC is given in fpecies. Take a ftraight line DE given in pofition and magnitude, and at the point D in the given ftraight line DE, make the angle EDF equal to the given angle BAC; wherefore the angle EDF is given; and because the ftraight line FD is drawn to the given point D in ED which is given in pofition, making the given angle EDF; therefore FD is given in pofition a. And because the ratio of BA to AC is given, make the ratio of ED to DF the fame with it, and join EF; and because the ratio of ED to DF is gi- B 4. 6. I. def. 41. 6. A a 32. dat. D C E F ven, and ED is given, therefore b DF is given in magnitude; b 2. dat. and it is given alfo in pofition, and the point D is given, wherefore the point F is given c; and the points D, E are given, e 30. dat. wherefore DE, EF, FD are given din magnitude; and thed 29. dat. triangle DEF is therefore given e in fpecies; and because thee 42. dat. triangles ABC, DEF have one angle BAC equal to one angle EDF, and the fides about thefe angles proportionals; the triangles are f fimilar; but the triangle DEF is given in fpecies, f 6. 6. and therefore alfo the triangle ABC. PROP. |