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are given therefore FB, the excess of AB above a given magnitude AF, has a given ratio to D."

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If two lines given in position cut one another, the point or points in which they cut one another are given.*

Let two lines AB, CD given in position cut one another in the point E; the point E is given.

C

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Because the lines AB, CD are given in position, they have always the same situation (4. def.), and therefore the point, or points, in which they cut one another, have always the same situation : and because the lines AB, CD can be found (4. def.), the point, or points, in which they cut one another, are likewise found; and therefore are given in position (4. def.).

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If the extremities of a straight line be given in position; the straight line is given in position and magnitude.

A

-B

Because the extremities of the straight line are given, they can be found (4. def.): let these be the points A, B, between which a straight line AB can be drawn (1. postulate.); this has an invariable position, because between two given points there can be drawn but one straight line: and when the straight line AB is drawn, its magnitude is at the same time exhibited, or given: therefore the straight line AB is given in position and magnitude.

3 C

See Note.

lels, BC, EF, the angle EAD is equal (29. 1.) 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 angle EAD, AD is given (32. dat.) in position.

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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 equal to it can be found (1 def.); 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

D.

E

be less than AE. If therefore D be equal to B AE, AE is the straight line given in magnitude, drawn from the given point A to BC: and it is evident that AE is given in position, (33. dat.), 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 position (6. def.), and the straight line BC is also given in position; therefore their intersection G is given (28.

A

dat.); and the point A is given;

wherefore AG is given in posi

tion (29. dat.), that is, the straight B GE

line AG given in magnitude, (for

it is equal to D) and drawn from

the given point A to the straight

line BC given in position, is also

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

*See Note.

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.

A

EH B

In CD take the given point G, and through G draw (31. 1.) GH parallel to EF: and because CD meets the parallels GH, EF, the angle EFD is equal (29. 1.) to the angle HGD: and EFD is 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 an- C gle HGD: the straight line HG is given in position (32. dat.): and AB is given in position: therefore the point H is given (28. dat.), and the point G is also given, wherefore GH is given in magnitude (29. dat.): and EF is equal to it, therefore EF is given in magnitude.

PROP. XXXVI.

F G

33.

If a straight line given in magnitude be drawn between 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 AEF, EFC shall be given.

Because EF is given in magnitude, a straight line equal to it can be found (1. def.) let this be G: in AB take a given point H, and from it draw (12. 1.) HK per- C pendicular to CD; therefore the straight

ЕН В

F K D

G

* See Note.

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 (6. def.) MLN, and the straight line CD, are given in position, the points M, N are (28. dat.) given and the point II is given, wherefore the straight A lines HM, HN, are given in position (29. dat.): and CD is given in position: therefore the angles HMN, HNM, are given

in position (A. def.): of the straight lines HM, HN, let HN be that which is not parallel to EF, for EF cannot be parallel

E

H

B

K

F

OM

D

G.

to both of them; and draw EO parallel to HN: EO therefore is equal (34. 1.) 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, (29. 1.), 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 magnitude (1. def.); and consequently the angle EFC.

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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.*

AH

F

Let the straight line AD given in magnitude be drawn from the point A to the straight line BC given in position, in the given angle ADC: the E 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

* Sce Note.

BD G

C

line BC given in position, in a given angle HGC, for it is equal (29. 1.) to the given angle ADC; HG is given in position (32. dat.); but it is given also in magnitude, because it is equal to (34. 1.) 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 given (30. dat.); and 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.

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Ir 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; EK is given

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in position (33. dat.); and AB, CD are given in position: therefore (28. dat.) the points H, K are given; and the point E is given; wherefore (29. dat.) EH, EK are given in magnitude, and the ratio (1. dat.) 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.

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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.

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