PROP. XXXII. Ir a straight line be drawn to a given point in a tion. Because the angle is given, one equal to it can be found; A let this be the angle at D. At the given point C, in the given straight line AB, make the angle ECB equal to the angle at G F 29. E F D: Therefore the straight line EC has always the same situation, because any other straight line FC, drawn to the point C, makes with CB a greater or less 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 observed, that there are two straight lines EC, GC, upon one side of AB that make equal angles with it, and which make equal angles with it when produced to the other side. b23. 1. PROP. XXXIII. If a straight line be drawn from a given point to a straight line given in position, and make a given angle with it, that straight line is given in position. From the given point A let the straight line AD be drawn to the straight line BC given in position, and make with it a given angle ADC: AD E is given in position. Through the point A draw a the straight line EAF parallel to BC; and because through the D F given point A the straight line EAF is drawn parallel to BC, which is given in position, EAF is therefore given in 30. a 31. 1. position. And because the straight line AD meets the paral- 31 Dat. 29. 1. lels BC, EF, the angle EAD is equal to the angle ADC; and ADC is 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 position, 32 Dat, and makes with it a given angle EAD, AD is given in position. PROP. XXXIV. See N. IF from a given point to a straight line given in position, a straight line be drawn which is given in magnitude; the same is also given in position. Let A be a given point, and BC a straight line given in position, a straight line given in magnitude, drawn from the point A to BC, is given in position. A Because the straight line is given in magnitude, one 1 Def. equal to it can be found a; let this be the straight line D: From the point A draw AE perpendicular to BC: and because AE is the shortest of all the straight lines which can be drawn from the point A to BC, the straight line D, to which one equal is to be drawn from the point A to BC, cannot be less than AE. If therefore D be equal to AE, AE is the straight line given in magnitude drawn from the given point A to BC: And it is evident that AE b 33 Dat. is given in position, because it is drawn from the given point A to BC, which is given in position, and makes with BC the given angle AEC. B D 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 centre A, at the distance AF, describe the circle GFH, and join AG, AH: Because the circle • 6 Def. GFH is given in position, and the straight line BC is also 28 Dat. given in position; therefore their intersection G is given; and the point A is given; wherefore AG is given inte 29 Dat. positione; that is, the straight line AG given in magnitude. (for it is equal to D), and drawn from the given point B G E H C I buspod DF A to the straight line BC given in position, is also given in position; And in like manner AH is given in position: Therefore in this case there are two straight lines AG, AH, of the same given magnitude which can be drawn from a given point A to a straight line BC given in position. PROP. XXXV. Ir a straight line be drawn between two parallel straight lines given in position, and makes given angles with them, the straight line is given in magnitade. Let the straight line EF be drawn between the parallels AB, CD, which are given in position, and make the given angles BEF, EFD: EF is given in magnitude. 32. a b In CD take the given point G, and through G draw a a 31. 1. GH parallel to EF: And because CD meets the parallels. GH, EF, the angle EFD is equal to the angle HGD: 29. 1. And EFD is a given angle; wherefore the angle HGD is given: and because HG is drawn to the given ЕН В point G, in the straight line CD, given in position, and makes a given angle HGD; the straight line HG FG 32 Dat. is given in position: And AB is given in position: therefore the point H is given: and the point G is also given, a 28. Dat. wherefore GH is given in magnitude: And EF is equal 29 Dat. to it, therefore EF is given in magnitude. If a straight line given in magnitude be drawn be- See N. tween two parallel straight lines given in position, it shall make given angles with the parallels. A EH B Let the straight line EF given in magnitude be drawn between the parallel straight lines AB, CD, which are given in position: The angles AEF, EFC, shall be given. a 1 Def. FK D G 12. 1. Because EF is given in magnitude, a straight line equal to it can be founda; Let this be G: In AB take a given point H, and from it drawb HK perpendicular to CD: Therefore the straight line G, that is, EF cannot be less than HK: And if G be equal to HK, EF also 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 absurd. Therefore the angle EFD is a right, and consequently 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 centre H, at the distance HL, describe the circle MLN, and join HM, HN: And because the • 6-Def. circle© MLN, and the straight line CD, are given in posia 28 Dat. tion, the points M, N ared given: And the point H is given, wherefore the straight lines HM, HN, are given in C F H B K ND • 29 Dat. positione: And CDis given in position; therefore the angles HMN, HNM, are given f A Def. in positionf: Of the straight lines HM, HN, let HN be that which is not parallel to EF, for EF cannot be parallel to both of them: and draw 34. 1. EO parallel to HN: EO therefore is equals to HN, that is, G to G; and EF is equal to G; wherefore EO is equal to 29. 1. EF, and the angle EFO to the angle EOF, that is, 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 EFD, that is, the angle AEF, is given in * 1 Def. magnitude: and consequently the angle EFC. See N. IF a straight line given in magnitude be drawn from a point to a straight line given in position, in a given angle; the straight line drawn through that point parallel to the straight line given in position, is given in position. E HE Let the straight line AD given in magnitude be drawn In BC take a given point G, and draw B DG C is drawn to a given point G in the straight line BC b given in position, in a given angle HGC, for it is equal to ▪ 29. 1. the given angle ADC; HG is given in position: But it is 32 Dat. given also in magnitude, because it is equal to AD, which © 34. 1. is given in magnitude: therefore because G, one of the extremities of the straight line GH, given in position, and magnitude, is given, the other extremity H is given; and a 30 Dat. the straight line EAF, which is drawn through the given point H parallel to BC given in position, is therefore given** 31 Dat. 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; 31. CG K b c EK is given in position; and AB, CD, are given in posi-a 33 Dat. tion; therefore the points H, K are given: And the point 28 Dat. E is given; wherefore EH, EK are given in magnitude, 29 Dat. and the ratio of them is therefore given. But as EH to d 1 Dat. EK, so is EF to EG, because AB, CD, are parallels; therefore the ratio of EF to EG is given. PROP. XXXIX. 35, 36. If the ratio of the segments of a straight line be. See N. tween a given point in it and two parallel straight lines, be given, if one of the parallels be given in position, the other is also given in position. |