IF a ftraight line be drawn to a given point in a given ftraight line, and makes a given angle with it. that ftraight line is given in pofition. Let AB be a straight line given in pofition, and C a given point in it, the ftraight line drawn to C which makes a given angle with CB, is given in pofition. Because the angle is given, one equal to it can be found; let this be the angle at D. at the given point C in the given ftraight line AB A make the angle ECB equal to the angle at D. therefore the ftraight line EC has always the fame fituati 29. F G E F B b. 23. 1. D on, because any other ftraight line FC drawn to the point C makes with CB a greater or lefs angle than the angle ECB or the angle at D. therefore the straight line EC which has been found is given in position. It is to be obferved that there are two ftraight lines EC, GC upon one fide of AB that make equal angles with it, and which make equal angles with it when produced to the other fide. IF PRO P. XXXIII. Fa ftraight line be drawn from a given point, to a ftraight line given in pofition, and makes a given angle with it; that straight line is given in pofition. From the given point A let the ftraight line AD be drawn to the ftraight line BC given in pofition, and make with it a given angle ADC; AD is given in pofition. 30. E Thro' the point A draw the ftraight A line EAF parallel to BC; and because thro' the given point A the ftraight line EAF is drawn parallel to BC which is gi- B ven in pofition, EAF is therefore given in pofition. and because the straight line AD meets the parallels BC, b. 31. Dat. c EF, the angle EAD is equal to the angle ADC; and ADC is e. 29. 1. given, wherefore also the angle EAD is given. therefore because the straight line DA is drawn to the given point A in the straight line EF given in pofition, and makes with it a given angle EAD; d. 32. Dat. AD is given in pofition. 31. See N. IF d PROP. XXXIV. F from a given point to a straight line given in position, a straight line be drawn which is given in magnitude; the fame is alfo given in pofition. Let A be a given point, and BC a straight line given in position, a ftraight line given in magnitude drawn from the point A to BC is given in pofition. Because the straight line is given in magnitude, one equal to it a. 1. Def. can be found; let this be the straight line D. from the point A draw AE perpendicular to BC; and because lefs than AE. If therefore D be equal to AE, A B E C D AE is the straight line given in magnitude drawn from the given b. 33. Dat. point A to BC. and it is evident that AE is given in pofition because it is drawn from the given point A to BC which is given in pofition, and makes with BC the given angle AEC. But if the straight line D be not equal to AE, it must be greater than it. produce AE, and make AF equal to D; and from the center A, at the distance AF defcribe the circle GFH, and join AG, c. 6. Def. AH, because the circle GFH is given in pofition, and the ftraight line BC is alfo given in pofition; therefore their interfection G is gi d. 28. Dat. vend; and the point A is given; BG E e. 19. Dat. wherefore AG is given in pofitione, that is, the ftraight line AG given in нс F D magnitude (for it is equal to D) and of of the fame given magnitude which can be drawn from a given point A to a straight line BC given in position. PROP. XXXV. 32. IF a ftraight line be drawn between two parallel ftraight lines given in pofition, and makes given angles with them; the ftraight line is given in magnitude. Let the straight line EF be drawn between the parallels AB, CD which are given in pofition, and make the given angles BEF, EFD ; EF is given in magnitude. A a EH B b. 29. I. In CD take the given point G, and thro' G draw GH parallel to a. 31. 1. EF. and because CD meets the parallels GH, EF, the angle EFD is equal to the angle HGD. and EFD is a given angle, wherefore the angle HGD is given. and becaufe HG is drawn to the given point G in the ftraight line CD given in pofition, and makes a given angle HGD; the ftraight line HG is given in pofition. and AB is given in pofition, therefore the point H is c. 32. Dat. given; and the point G is alfo given, wherefore GH is given in d. 28. Dat. magnitude, and EF is equal to it; therefore EF is given in mag- e. 29. Dat. nitude. 譬 1 F G D IF PROP. XXXVI. 33. Fa ftraight line given in magnitude be drawn between Sec N. two parallel ftraight lines given in pofition; it fhall make given angles with the parallels. Let the ftraight line EF given in magnitude be drawn between the parallel straight lines AB, CD which are given in pofition; the angles AEF, EFC shall be given. A C EHB a. I. Def. FK D Becaufe EF is given in magnitude, a ftraight line equal to it can be found"; let this be G. in AB take a given point H, and from it draw b HK perpendicular to CD. therefore the straight line G, that is EF, cannot be lefs than HK. and and if G be equal to HK, EF alfo is equal to it; wherefore EF is at right angles to CD, for if it be not, EF would be greater than HK, which is abfurd. therefore the angle EFD is a right and confe quently a given angle. But if the straight line G be not equal to HK, it must be greater than it. produce HK, and take HL equal to G; and from the center H, at the distance HL describe the circle MLN, and join HM, c. 6. Def. HN. and because the circle MLN, and the straight line CD are d. 28. Dat. given in pofition, the points M, N are given; and the point H is c given, wherefore the straight lines HM, HN are given in po c. 29. Dat. fition, and CD is given in pofition, therefore the angles HMN, f. A. Def. HNM are given in position f. of d the ftraight lines HM, HN let lel to EF, for EF cannot be pa rallel to both of them; and draw EO parallel to HN. EO there34.1. fore is equal to HN, that is to G; and EF is equal to G, where fore EO is equal to EF, and the angle EFO to the angle EOF, that h. 29. 1. ish to the given angle HNM. and because the angle HNM which is equal to the angle EFO or EFD has been found, therefore the angle k. 1. Def. EFD, that is the angle AEF, is given in magnitude k, and confequently the angle EFC. E. See N. PROP. XXXVII. IF a straight line given in magnitude be drawn from a point to a straight line given in pofition, in a given angle; the ftraight line drawn thro' that point parallel to the ftraight line given in pofition, is given in pofition. Let the straight line AD given in magnitude be drawn from the point A to the straight line BC given in pofition, in the given angle ADC; the straight line E A H F EAF drawn thro' A parallel to BC is given in pofition. In BC take a given point G, and draw GHB D G C parallel to AD. and because HG is drawn to a given point G in the ftraight line BC given in pofition, in a given angle ngle HGC, for it is equal to the given angle ADC; HG is given a. 29. 1. in pofition ; but it is given alfo in magnitude, because it is equal to b. 32. Dat. AD which is given in magnitude. therefore because G one of the extremities of the straight line GH given in pofition and magnitude is given, the other extremity H is given. and the straight line c. 30. Dat. EAF which is drawn thro' the given point H parallel to BC given in pofition, is therefore given in position. IF a d PRO P. XXXVIII. a ftraight line be drawn from a given point to two parallel ftraight lines given in pofition; 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 becaufe from a given point E the straight line EK is drawn to CD which is given in pofition, in a given angle EKC; EK is given in posi d. 31. Dat. 34. b. 28. Dat C. 29. Dat. tion. and AB, CD are given in pofition; therefore the points a. 3. Dafi H, K are given. and the point E is given, wherefore EH, EK are given in magnitude, and the ratio of them is therefore given. but as EH to EK, fo is EF to EG, because AB, CD are parallels. therefore the ratio of EF to EG is given. PROP. XXXIX. d. 1. Dat. 35.36: IF the ratio of the fegments of a straight line between a See N. given point in it and two parallel ftraight lines given in position, be given; if one of the parallels be given in pofition, the other is alfo given in pofition. |