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c 29. 1. BC, EF, the angle EADc 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, and makes with it a given d 32. dat. angle EAD, AD is givend in position.

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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.,

Because the straight line is given in magnitude, one equal to a 1. def. it can be founda; let this be the straight line D: from the point A draw AE perpendicular to BC; and

A 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 B E C D be equal to AE, AE is the straight line D

given in magnitude drawn from the given point A to BC: and b 33. dat. it is evident that AE is given in position b, because it is drawn

from the given point A to BC, which is given in position, 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 centre A, at the distance AF, describe the circle GFH,

and join AG, AH : because the circle GFH is given in posic 6. def. tionc, and the straight line BC is also given in position ; therefore their intersection G is gi

A d 28. dat. vend; and the point A is gi

ven; wherefore AG is given in e 29. dat. positione, that is, the straight

H

B
line AG given in magnitude,
(for it is equal to D) and drawn
from the given point A to the

F
straight line BC given in posi-

Dtion, is also given in position: and in like manner AH is given in position: therefore in this case there are two straight lines

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G E

AG, AH of the same given magnitude which can be drawn from a given point A to a straight line BC given in position.

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IF 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 magnitude.

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.

In CD take the given point G, and through G draw a GH a 31. 1. parallel to EF: and because CD meets the parallels GH, EF, the angle EFD is equal b to the angle

b 29.1.

A E H HGD: and EFD is a given angle ;

B wherefore the angle HGDis given; and because HG is drawn to the given point G, in the straight line CD, given in position, and makes a given angle HGD:

C F G

D the straight line HG is given in position c: and AB is given in position : therefore the point H is c 32. dat. givend; and the point G is also given, wherefore GH is given d 28. clat in magnitude e: and EF is equal to it, therefore EF is given in e 29, dat. magnitude.

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IF a straight line given in magnitude be drawn be-Sec N. tween two parallel straight lines given in position, it shall make given angles with the parallels

Let the straight line EF given in magnitude be drawn between the parallel straight lines AB, CD, which are given in position : the angles A E_IB AEF, EFC shall be given.

Because EF is given in magnitude, a straight line equal to it can be found a :

a 1. def. let this be G : in AB take a given point_ H, and from it draw b HK perpendicu-O F K Db 12.1. lar 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 c 6. def. MLN, and join HM, HN: and because the circlec MLN, and

the straight line CD, are given in position, the points M, N are d 28. dat. given: and the point His given, wherefore the straight A E

H

B lines HM, HN, are given e 29. dat. in position e: and CD is

given in position; therefore
the angles HMN, HNM,

C
OM

N D f A. def. are given in position f:of the

straight lines HM, HN, let
HN be that which is not pa G-
rallel to EF, for EF cannot
be parallel to both of them; and draw EO parallel to HN: EO
therefore is equal 8 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 h 29. 1.

EOF, that is h, 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 k 1. def. given in magnitudek; and consequently the angle EFC.

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& 34. 1.

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See Note.

IF a straight line given in magnitude be drawn from a point to a straight line given in poi ion, in a given angle; the straight line drawn through that poin: parallel to the straight line given in posi ion, is given in position.

Let the straight line AD given in magnitude be drawn from the point A to the straight line PC given in position, in the given angle ADC: the F Α. Η F straight line EAF drawn through A parallel to BC is given in position.

In BC take a given point G, and draw GH parallel to AD: and because HG is drawn to a given point G in the straight

B D G

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line BC given in position, in a given angle HGC, for it is equal a 29. 1. a to the given angle ADC; HG is given in position b; but it is b 32. dat. given also in magnitude, because it is equal to C AD which is c 34. 1. 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 d; and the straight line d 30. dat. EAF, which is drawn through the given point H parallel to WC given in position, is therefore given e in position.

e 31 dat.

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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; F.K is

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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 c EH, EK are given in magnitude, and the ratio d of c 29. dat. them is therefore given. But as EH to EK, so is EF to EG, d 1. dat. because AB, CD are parallels ; therefore the ratio of EF to LG 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.

From the given point A, let the straight line AED be draw 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 position, the other FG is also given in position.

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

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B

H D C
B
D H

C
a 33. dat. given angle AHD; AH is given a in

position; and BC is likewise given in

position, therefore the point H is giv- B D H с b 28. dat. en b: the point A is also given ; wherec 29. dat. fore AH is given in magnitude c, and,

because FG, BC are parallels, as AE
to AD, so is AK to AH; and the

F E KG ratio of AE to AD is given, where

fore the ratio of AK to AH is given ; but AH is given in magd 2. dat. nitude, therefore d AK is given in magnitude ; and it is also e 30. dat. given in position, and the point A is given; whereforee the point

K is given. And because the straight line FG is drawn through

the given point K parallel to PC which is given in position f31. dat. therefore f F G is given in position.

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See N.

IF the ratio of the segments of a straight line inte which it is cut by three parallel straight lines, be giv. en; if two of the parallels are given in position, the third also is given in position.

Let AB, CD, HK be three parallel straight lines, of which AB, CD are given in position ; and let the ratio of the seg.

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