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* 12. 1.

* 6 Def.

between the parallel straight lines
AB, CD, which are given in po-

A Е н в
sition: the angles XEF, EFC,
shall be given

Because EF is given in magni

tude, a straight line equal to it can С F K D * 1 Def, be found *; let this be G: in AB

G take a given point H, and from it draw* 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 circle * MLN, and the straight line * 28 Dat. CD, are giveri in position, the points M, N given:

and the point His
given, wherefore the A

H Н B
straight lines HM, HN,
are given in position*:
and CD is given in

IK position ; therefore the C F OM ND

angles HMN, HNM, * A. Def. are given in position *;

G of the straight lines HM, HN, let. #N be that which is not parallel to EF, for

EF cannot be parallel to both of them : and draw EO * 34. 1.

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 because the angle HNM, which is equal to the angle EFO, or EFD,

has been found; therefore the angle EFD, that is, the * 1 Def. angle AEF, is given in magnitude *; and consequently

the angle EFC.

* are

* 29 Dat.

* 29. 1.

E.

PROP. XXXVII.

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

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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 position, E A HF in the given angle ADC; the straight line EAF drawn through A parallel to BC is given in position.

In BC take a given point G, and B D G C draw GH 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 HGC, for it is equal * to the given angle ADC; HG is given in

29. 1.' position *: but it is given also in magnitude, because it . 32 Dat. is equal to * AD, which is given in magnitude: there- * 34. 1. fore 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 the straight line * 30 Dat. EAF, which is drawn through the given point H parallel to BC given in position, is therefore given * in * 31 Dat. position.

PROP. XXXVIII.
If a straight line be drawn from a given point to two pa-

rallel 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; EK is given in position *; and AB, CD, * 33 Dat. E

А. FH B

34.

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A

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C G

K

D

are given in position ; therefore * the points H, K * 28 Dat.

• 29 Dat. * 1 Dat.

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, so is EF to EG, because AB, CD, are parallels; therefore the ratio of EF to EG is given.

PROP. XXXIX.

35, 36.

See N.

If the ratio of the segments of a straight line between 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 drawn 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 given angle AHD; AH is

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• 28 Dat.

• 29 Dat.

B D H с
• 33 Dat. given * in position ; and BC

is likewise given in position,
therefore the point His given *
the point A is also given;

where- B DUH
fore AH is given in magni-
tude *, and, because FG, BC,
are parallels, as AE to AD, so
is AK to AH; and the ratio F E K G
of AE to AD is given, where-

fore the ratio of XK to AH is given; but AH is given * 2 Dat. in magnitude, therefore * AK is given in magnitude; and it is also given in position, and the point

A is given : wherefore the point K is given. And because the straight line FG is drawn through the given point K

30 Dat.

parallel to BC which is given in position, therefore * * 31 Dat. FG is given in position.

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If the ratio of the segments of a straight line into which See N.

it is cut by three parallel straight lines be given ; if two of the parallels be given in position, the third is also 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 segments GE, GF, into which the straight line GEF is cut by the three parallels, be given; the third parallel HK is given in position.

In AB take a given point L, and draw LM perpendicular to CD, meeting HK in N: because LM is drawn from the given point L to CD which is given in position, and makes a given angle LMD; LM is given in position *; and CD is given in position, where- * 33 Dat. fore the point M is given*; and the point L is given; LM * 28 Dat. is therefore given in magnitude* ; and because the ratio • 29 Dat. of GE to GF is given, and as GE to GF, so is NL to H G N K

E L B

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D

* Cor

CF M

MD NM; the ratio of NL to NM is given; and therefore * * Cor.6. the ratio of ML to LN is given; but ML is given in or Dat. magnitude *, wherefore * LN is given in magnitude : or 7 Dat. and it is also given in position, and the point L is given, * 2 Dat. wherefore * the point N is given : and because the • 30 Dat. straight line HK is drawn through the given point N parallel to CD, which is given in position, therefore HK is given in position *.

* 31 Dat.

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If a straight line meet three parallel straight lines which See N.

are given in position; the segments into which they cut
it have a given ratio.
Let the parallel straight lines AB, CD, EF, given

* 33 Dat.

39.

See N.

in position, be cut by the straight line GHK; the ratio. of GH to HK is given.

In AB take a given point L, and draw LM perpendicular to

A G L B CD, meeting EF in N; therefore * LM is given in position :

CH, M D and CD, EF, are given in position, wherefore the points M, N, are given : and the point L is

N F * 29 Dat. given; therefore * the straight EK

lines LM, MN, are given in

magnitude; and the ratio of LM to MN is therefore 1 Dat. given *: but as LM to MN, so is GH to HK; wherefore the ratio of GH to HK is given.

PROP. XLII.
If each of the sides of a triangle be given in magnitude

the triangle is given in species.
Let each of the sides of the triangle ABC be given

in magnitude, the triangle ABC is given in species. * 22. 1. Make a triangle* DEF, the sides of which are

equal, each to each, to the given straight lines AB,
BC, CA; which can be done, because any two of them
must be greater than
the third ; and let

A

D DE be equal to AB, EF to BC, and FD to CA, and because the two sides ED, DF, are equal to the B

F two BA, AC, each to each, and the base EF equal to the base BC; the angle EDF is equal * to the angle BAC; therefore, be

cause the angle EDF, which is equal to the angle *1 Def. BAC, has been found, the angle BAC is given * : in

like manner the angles at B, C, are given. And because the sides AB, BC, CA, are given, their ratios to

one another are given *; therefore the triangle ABC is 3 Def. given * in species.

PROP. XLIII.
If each of the angles of a triangle be given in magnitude ;

the triangle is given in species.
Let each of the angles of the triangle ABC be given
in magnitude, the triangle ABC is given in species.

44

* 8. 1.

* 1 Dat.

40.

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