! 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 CAD which is given inc 34. 1. magnitude; therefore because G one of the extremities of the ftraight line GH given in position and magnitude is given, the other extremity H is given d; and the straight line EAF, which d 30. dat. is drawn through the given point H parallel to BC given in pofition, is therefore given e in position. PROP. XXXVIII. IF a straight line be drawn from a given point to two parallel straight lines given in position; the ratio of the segments between the given point and the parallels shall be given. Let the straight 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 straight line EK is drawn to CD which is given in position, in a given angle EKC; EK is E AFH B A FH IB e 31. dat. 34 CGK C KG D D given in position a; and AB, CD are given in position; there a 33. dat. fore b the points H, K are given : And the point E is given; b 28. dat. wherefore EH, EK are given in magnitude, and the ratio d ofc 29. dat. them is therefore given. But as EH to EK, so is EF to EG, be. d 1. dat. cause AB, CD are parallels; therefore the ratio of EF to EG is given. PROP. XXXIX. 35. 36. IF the ratio of the fegments of a ftraight line between Sce 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 also given in pofition. From From the given point A, let the straight line AED be drawn to the two parallel straight lines FG, BC, and let the ratio of the segments AE, AD be given; if one of the parallels BC be given in pofition, the other FG is also 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 straight line BC given in position, and makes a ■ 33. dat. given angle AHD; AH is given a in pofition; and BC is likewife given in position, therefore the point H is gi- B A DHC b 28. dat. ven b: The point A is also given; C 29. dat. wherefore AH is given in magnitude c, and, because FG, BC are parallels, as AE to AD, so is AK to AH; and the ratio of AE to AD is given, FE wherefore the ratio of AK to AH is given; but AH is given KG 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 straight 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 segments of a straight line into which it is cut by three parallel straight lines, be given; if two of the parallels are given in position, the third also is given in pofition. Let AB, CD, HK be three parallel straight lines, of which AB, CD are given in position; and let the ratio of the seg ments ments GE, GF into which the straight line GEF is cut by the three parallels, be given; the third parallel HK is given in posi tion. In AB take a given point L, and draw LM perpendicular to CD, meeting HK in N; because LM is drawn from the given point L to CD which is given in position and makes a given angle LMD; LM is given in positiona; and CD is given a 33. dat. in position, wherefore the point Mis given b; and the point Lb 28. dat. is given, LM is therefore given in magnitude; and becausec 29. dat. the ratio of GE to GF is given, and as GE to GF, so is NL to Cor. 6. or 7. dat. e 2. dat. NM; the ration of NL to NM is given; and therefored the ratio of ML to LN is given; but LM is given in magnitude, d wherefore e LN is given in magnitude; and it is also given in pofition, and the point L is given; wherefore f the point N is f 30. dat. given; and because the straight line HK is drawn through the given point N parallel to CD which is given in position, therefore HK is given in position g. PROP. XLI. g 31. dat. F. IF a straight line meets three parallel straight lines See N. which are given in position; the segments into which they cut it have a given ratio. Let the parallel straight lines AB, CD, EF given in pofition be cut by the straight line GHK; the ratio of GH to HK is given. In AB take a given point L, and draw LM perpendicular to CD, meet- A ing EF in N; therefore a LM is given GLB a 33, dat. D H/M in pofition; and CD, EF are given CHM in position, wherefore the points M, Nare given: And the point L is given; therefore b the straight lines LM, MN are given in magnitude; and the ratio EK. NF b 29. data of c. 1. dat. of LM to MN is therefore given c: But as LM to MN, so is GH to HK; wherefore the ratio of GH to HK is given. Sce N. IF each of the fides of a triangle be given in magnitude; the triangle is given in species. Let each of the fides of the triangle ABC be given in magnitude; the triangle ABC is given in species. 22. 1. Make a triangle a DEF the fides of which are equal, each to each, to the given straight lines AB, BC, CA; which can be done, because any two of them must be greater than the third; and let DE be e qual to AB, EF to BC, D and FD to CA; and be cause the two fides ED, DF are equal to the two BA, AC, each to each, and the base EF equal to B CE F b 8. 1. the bafe BC; the angle EDF is equal b to the angle BAC; therefore, because the angle EDF, which is equal to the angle BAC, has been found, the c. 1. def. angle BAC is given c, in like manner the angles at B, Care given. And because the fides AB, BC, CA are given, their d. 1. dat. ratios to one another are given d, therefore the triangle ABC is given e in species. 4. 3. def. IF each of the angles of a triangle be given in magni tude; the triangle is given in species. Let each of the angles of the triangle ABC be given in mag nitude; the triangle ABC is given in species. Take a straight line DE given in 2 23. 1. pofition and magnitude, and at the angle DEF equal to ABC; there fore the other angles EFD, BСА В are equal; and each of the angles at the points A, B, C, is gi ven 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 position, making the given angle EDF; therefore DF is given in position b. In like manner EF b 32. dat. also is given in position; wherefore the point F is given: And the points D, E are given; therefore each of the straight lines DE, EF, FD is given e in magnitude; wherefore the triangle c 29. dat. DEF is given in species d; and it is fimilare to the triangled 42. dat. ABC; which therefore is given in species. PROP. XLIV. IF 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 species. 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 species. Take a straight line DE given in position and magnitude, and at the point D in the given straight line DE, make the angle EDF equal to the given angle BAC; wherefore the angie EDF is given; and because the straight line FD is drawn to the given point D in ED which is given in position, making 4. б. 41. the given angle EDF; therefore FD is given in pofition a. And because a 32. dat the ratio of BA to AC is given, make the ratio of ED to DF the D fame with it, and join EF; and be cause the ratio of ED to DF is gi- B CEF 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, where fore the point F is given c; and the points D, E are given, 30. dat. wherefore DE, EF, FD are given din magnitude; and thed 29. dat. triangle DEF is therefore given ein species; and because thee 42. dat. triangles ABC, DEF have one angle BAC equal to one angle EDF, and the fides about these angles proportionals; the triangles are f fimilar; but the triangle DEF is given in species, f 6. 6. and therefore also the triangle ABC. PROP. |