29.

F E

a. 1. Def.

PRO P. XXXII.
IF
F a straight line be drawn to a given point in a given

straight line, and makes a given angle with it. that straight line is given in polition.

Let AB be a straight line given in position, and C a given point in it, the straight line drawn to C which makes a given angle with CB, is given in position.

G · Because the angle is given, one

F 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 A

B make b the angle ECB equal to the angle at 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.

b. 23. 1.

P RO P. XXXIII.

30. IF F a straight line be drawn from a given point, to a

straight line given in position, and makes 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 is given in position.
Thro' the point A draw the straight

E A F line EAF parallel to BC; and because thro' the given point A the straight line EAF is drawn parallel to BC which is gi- B D C ven in position, EAF is therefore given in position b, and because the straight line AD meets the parallels BC, b. 31. Dat. EF, the angle EAD is equal o to the angle ADC; and ADC is c. 29. 1.

2. 31. I.

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 angle EAD; d. 32. Dat. AD is given d in position.

31.

PRO P. XXXIV.

See N.

IF
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 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 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

A
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

B E C less than AE. If therefore D be equal to AE,

AE is the straight line given in magnitude drawn from the giren b. 33. Dat. point A to BC. and it is evident that AÉ is given in position b be

cause 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 cen

ter A, at the distance AF describe the circle GFH, and join AG, c. 6. Def. AH. because the circle GFH is given in position, and the straight line BC is also given in position ;

A therefore their interfection G is gid. 28. Dat. vend; and the point A is given; BG E H C 6. 29. Dat. wherefore AG is given in position, that is, the straight line AG given in

D magnitude (for it is equal to D) and drawn from the given point A to the straight line BC given in polition, is also given in pofition, and in like manner AH is given in pofition. 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.

PRO P.

XXXV.

32.

IF
F a straight line be drawn between two parallel straight

lines given in pofition, and makes given angles with them; the straight line is given in magnitude.

EH B 1.29. 1.

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 thro' G draw.GH parallel to 2. 31. 5. EF. and because CD meets the parallels GH, EF, the angle EFD is equal to the angle HGD. and EFD is

A

b. a given angle, wherefore the angle HGD is given, and because HG is drawn to the given point G in the straight line CD given in position, and makes a given angle

с F G D HGD; the straight line HG is given in position, and AB is given in position, therefore the point H is c. 32. Dat. given d; and the point G is also given, wherefore GH is given in d. 28. Dat. magnitude and EP is equal to it; therefore EF is given in mag. c. 29. Dat. nitude.

33

PRO P. XXXVI. IF F a straight line given in magnitude be drawn between Sec N.

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

A Енв given in position; the angles AEF, EFC shall be given.

Because EF is given in magnitude, a straight line equal to it can be found ; let this be G. in AB take a given point H, and

C F K D from it draw 5 HK perpendicular to CD.

G therefore the straight line G, that is EF, cannot be less 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 absurd. therefore the angle EFD is a right and conse-
quently a given angle.

But if the straight line 6 be not equal to HK, it must be greater
than it. produce HK, and take HL equal to G; and from the cen-

ter H, at the distance HL defcribe 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 position, the points M, N are d given; and the point His

given, wherefore the straight
lines HM, HN are given in po-

A

E H c. 29. Dat. fition, and CD is given in pofi

tion, therefore the angles HMN, f. A. Def. HNM are given in position f. of

F
the straight lines HM, HN let

OM IND
HN be that which is not paral G
lel to EF, for EF cannot be pa-

rallcl to both of them; and draw EO parallel to HN. EO there3. 34. 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. is h to the given angle HNM. and because the angle ANM 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 magnitudek, and confe

quently the angle EFC.

E.

See N.

PRO P. XXXVII.
IF a straight line given in magnicude be drawn from a

point to a straight line given in position, in a given
angle; the straight line drawn thro’ that point parallel to
the straight line given in position, is given in position.

Let the straight line AD given in magnitude be drawn from the
point A to the straight line BC given in pofiti-
on, in the given angle ADC; the straight line

E A HF
EAF drawn thro' A parallel to BC is given in
position.
In BC take a given point G, and draw GH

B D G C
parallel to AD. and because HG is drawn to a
given point G in the straight line BC given in position, in a given

angle

engle HCC, for it is equal to the given angle ADC; HG is given a. 29. I. in position b; but it is given also 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 position and magnitude is given, the other extremity H is giveno. and the straight line c. 30. Dat. EAF which is drawn thro' the given point H parallel to BC given in position, is therefore given din position.

d. Data

31.

P R O P. XXXVIII.

34.

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 anġle EKC; EK is given in posid

E

A FH B
A F

H

B

С K G D C G K D tion'. and AB, CD are given in position; therefore the points a. 33. D." H, K are given. and the point E is given, wherefore EH, EK are b. 28. Dato given in magnitude, and the ratio of them is therefore given. but: 29. Dat

d. 1. Dat. as EH to EK, fo is EF to EG, because AB, CD are parallels. therefore the ratio of EF to EG is given.

PRO P. XXXIX.

35. 36.

IF the ratio of the segments of a straight line between a see N.

given point in it and two parallel straight lines given in position, be given; if one of the parallels be given in position, the other is also given in position.

Bb

From