PROP. XXXII. TF a ftraight line be drawn to a given point in a given ftraight line, and makes a given angle with it. that straight line is given in position. 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 make the angle ECB equal to the angle at D. therefore the ftraight line EC has always the fame fituation, because any other ftraight line 29. G FE F 2. 1. Def. A C B D 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 ftraight line IC which has been found is given in pofition. 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 a ftraight line be drawn from a given point, to a fraight line given in pofition, and makes a given angle with it; that ftraight 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. Thro' the point A draw the ftraight E A line EAF parallel to BC; and bccaufe 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 becaufe the ftraight line AD meets the parallels BC, b. 31. Dat. C EF, the angle EAD is equal to the angle ADC; and ADC is c. 29. 1. given, wherefore alfo the angle EAD is given. therefore beta the ftraight line DA is drawn to the given point A in the ftraigh line EF given in pofition, and makes with it a given angle EAD: d. 32. Dat. AD is given in pofition. See N. 31. IF F from a given point to a straight line given in pɔtion, 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 ia poltion; 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 tok a. 1. Def. can be found; let this be the ftraight line D. from the point å draw AE perpendicular to PC; and because AE is the fhorteft of all the ftraight lines C AE is the straight line given in magnitude drawn from the gha b. 33. Dat. point A to BC. and it is evident that AE is given in pofition be caufe it is drawn from the given point A to BC which is gira in pofition, and makes with BC the given angle AEC. But if the ftraight line D be not equal to AE, it must be greate than it. proface AE, and make AF equal to D; and from the c ter A, at the diftance AF defcribe the circle GFII, and join AG, c. 6. Def. AH. because the circle GFH is given in pofition, and the straight line BC is alfo given in pofition; d. 28. Dat. e. 29. Dat. A I C F D therefore their interfection G is gi- All of the fame given magnitude which can be drawn from a given point A to a ftraight line BC given in pofition. PROP. XXXV. Faftraight line be drawn between two parallel ftraight bas given in pofition, and makes given angles with them; the ftraight line is given in magnitude. Let the Ardht line EF be drawn between the parallels AB, C1) which given in pofition, and make the given angles BEF, LUD, ET is even in magnitude. A 32. b. 29.2. EH B In CD take the given point G, and thro' G draw GH paral- 2. 31. 1. lel to EF. and because CD) meets the parallels GHI, EF, the angle ED) is equd 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 e. 32. Dat. H is given; and the point G is alfo given, wherefore GH is given d. 28. Dat. in magnitude. and EF is equal to it; therefore EF is given ine. 29. Dat. magnitude. FG D PROP. XXXVI. 33. IF Fa ftraight line given in magnitude be drawn between See 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 ftraight lines AB, CD which are given in pofition; the angles AEF, EFC shall A C EHB therefore the ftraight line G, that is EF, cannot be lefs than HK. 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, EP would be greater than HK, which is abfurd. therefore the angle EFD is a right and confequently a given angle. But if the ftraight 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 diftance HL defcribe the circle MLN, and join c. 6. Def. HM, HN. and because the circle MLN, and the ftraight line CD d. 28. Dat. are given in pofition, the points M, N are given; and the point I c rallel to EF, for EF cannot be 8.34. I. h. 29. 1. g parallel to both of them; and draw EO parallel to HN. EO therefore is equal to HN, that is to G; and EF is equal to G, wherefore EO is equal to EF, and the angle EFO to the angle EOF, that is to the given angle HNM. and becaufe the angle HNM which is equal to the angle EFO or EFD has been found, therefore the angle EFD, that is the angle AEF, is given in magk. 1. Def. nitude k, and confequently the angle EFC. See N. E. PROP. XXXVH. IF a ftraight line given in magnitude be drawn from a point to a ftraight 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. E AHE Let the ftraight line AD given in magnitude be drawn from the point A to the ftraight line BC given in pofition, in the given angle ADC; the straight line EAF drawn through A parallel to BC is given in pofition. In BC take a given point G, and draw GH BD G. C parallel to AD. and because HG is drawn to a given point G in the straight line BC given in pofition, in a given angle HGC, for it is equal to the given angle ADC; HG is gi- a. 29. 1. ven in pofition; but it is given alfo in magnitude, because it is e- b. 31. Dat. qual to AD which is given in magnitude. therefore becaufe G one of the extremities of the ftraight line GH given in pofition and magnitude is given, the other extremity H is given. and the c. 30. Dat. ftraight line EAF which is drawn through the given point H parallel to BC given in pofition, is therefore given in pofition. IF PROP. XXXVIII. F 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 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 given in pofi d. 31. Dat 34. C c. 29. Dat. tion. and AB, CD are given in pofition; therefore the points a. 33. Dat. H, K are given. and the point E is given, wherefore EH, EK b. 28. Dat. are given in magnitude, and the ratio of them is therefore given. d. 1. Dat. bat 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. 35.36. F the ratio of the fegments of a straight line between See N. IF 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. Bb |