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given, wherefore each of those at the points D, E, F is given: And because the straight line FD is drawn to the given point D in DE, which is given in position, making the given angle EDF; therefore DF is given in position b. In like manner, b 32. dat. EF also is given in position; wherefore the point F is given : And the points D, E are given; therefore each of the straight lines DE, EF, FD is given in magnitude; wherefore the tri- c 29. dat. angle DEF is given in species d: and it is similar to the tri- d 42. dat. angle ABC; which therefore is given in species.

PROP. XLIV.

If one of the angles of a triangle be given, and if the sides about it have a given ratio to one another; the triangle is given in species.

Let the triangle ABC have one of its angles BAC given, and let the sides BA, AC about it have a given ratio to one another; the triangle ABC is given in species.

Take a straight line DE given in position and magnitude, and at the point D, in the given straight line DE, make the angle EDF equal to the given angle BAC; wherefore the angle EDF is given; and because the straight line FD is drawn to the given point D in ED, which is given in posi

e

41.

4.6.
1. def.
6.

tion, making the given angle EDF; therefore FD is given in

positiona. And because the ratio of BA to AC is given, make the ratio of ED to DF the same

with it, and join EF; and be- B

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cause the ratio of ED to DF is

given, and ED is given, therefore DF is given in magni- b 2. dat. tude: and it is given also in position, and the point D is

given, wherefore the point F is given and the points D, c 30. dat. E are given, wherefore DE, EF, FD are given in magni- d 29. dat. tude; and the triangle DEF is therefore given in species; e 42. dat. and because the triangles ABC, DEF have one angle BAC equal to one angle EDF, and the sides about these angles proportionals; the triangles are similar; but the triangle f. 6. 6. DEF is given in species, and therefore also the triangle

ABC.

See N.

42.

a 2. dat.

PROP. XLV.

If the sides of a triangle have to one another given ratios; the triangle is given in species.

Let the sides of the triangle ABC have given ratios to one another, the triangle ABC is given in species.

Take a straight line D given in magnitude; and because the ratio of AB to BC is given, make the ratio of D to E the same with it; and D is given, therefore a E is given. And because the ratio of BC to CA is given, to this make the ratio of E to F the same ; and E is given, and therefore F. And because as AB to BC, so is D to E; by composition, AB and BC together are to

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D, E, F, are greater than the

e 22. 1.

g 5. 6.

third. Make the triangle GHK having its sides equal to D, E, F, so that GH may be equal to D, HK to E, and KG to F; and because D, E, F, are each of them given, therefore GH, HK, KG are each of them given in magf 42. dat. nitude; therefore the triangle GHK is given in species: But as AB to BC, so is (D to E, that is) GH to HK; and as BC to CA, so is (E to F, that is) HK to KG; therefore, ex æquali, as AB to AC, so is GH to GK. Wherefore & the triangle ABC is equiangular and similar to the triangle GHK; and the triangle GHK is given in species; therefore also the triangle ABC is given in species.

COR. If a triangle is required to be made, the sides of which shall have the same ratios which three given straight lines D, E, F have to one another; it is necessary that every two of them be greater than the third,

PROP. XLVI.

If the sides of a right angled triangle about one of the acute angles have a given ratio to one another ; the triangle is given in species.

Let the sides AB, BC about the acute angle ABC of the triangle ABC which has a right angle at A, have a given ratio to one another; the triangle ABC is given in species.

43.

Take a straight line DE given in position and magnitude; and because the ratio of AB to BC is given, as AB to BC, so make DE to EF; and because DE has a given ratio to EF, and DE is given, therefore a EF is given; and because as AB a 2. dat. to BC, so is DE to EF; and AB is less than BC; therefore b 19. 1. DE is less than EF. From the point D draw DG at right an- c A.5.

gles to DE, and from the centre E, at the distance

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C

points; let G be either of them, and join EG; therefore the circumference of

F

F

G

the circle is given d in position; and the straight line DG is d 6. def. given in position, because it is drawn to the given point De 32. dat. in DE given in position, in a given angle; therefore the f 28. dat. point Gisgiven: and the points D, E are given, wherefore DE, EG, GD are given & in magnitude, and the triangle DEG in g 29. dat. species h. And because the triangles ABC, DEG have the h 42. dat. angle BAC equal to the angle EDG, and the sides about the angles ABC, DEG proportionals, and each of the other angles BCA, EGD less than a right angle; the triangle ABC is equiangular and similar to the triangle DEG: But DEG i 7. 6. is given in species; therefore the triangle ABC is given in species: And in the same manner, the triangle made by drawing a straight line from E to the other point in which the circle meets DG is given in species.

44.

See N.

a 32. 1.

b 43. dat.

PROP. XLVII.

If a triangle has one of its angles which is not a right angle given, and if the sides about another angle have a given ratio to one another; the triangle is given in species.

Let the triangle ABC have one of its angles ABC a given angle, but not a right angle, and let the sides BA, AC about another angle BAC have a given ratio to one another; the triangle ABC is given in species.

First, Let the given ratio be the ratio of equality, that is, let the sides BA, AC, and consequently the angles ABC, ACB be equal; and because the angle ABC is given, the angle ACB, and also the remaining a angle BAC is given; therefore the triangle ABC is given in species; and it is evident, that, in this case, the given angle ABC must be

acute.

B

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Next, Let the given ratio be the ratio of a less to a greater, that is, let the side AB adjacent to the given angle be less than the side AC: Takeastraight line DE, given in position and magnitude, and make the angle

c 32. dat. DEF equal to the given angle ABC; therefore EF is given c in position; and because the ratio of BA to AC is given, as BA to AC, so make ED to DG; and because the ratio of

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g 28. dat. point Fis given ; and the points D, E are given; wherefore the

straight lines DE, EF, FD are

G

A

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h 29. dat. given hin magnitude, and the triangle DEF, in species. And i 42. dat. because BA is less than AC, the angle ACB is less than the k 18. 1. angle ABC, and therefore ACB is less than a right angle.

1 17.1.

In the same manner, because ED is less than DG or DF, the angle DFE is less than a right angle: And because the triangles ABC, DEF have the angle ABC equal to the angle DEF, and the sides about the angles BAC, EDF proportionals, and each of the other angles ACB, DFE less than a right angle; the triangles ABC, DEF are similar, and DEF is given in m 7. 6. species, wherefore the triangle ABC is also given in species. Thirdly, Let the given ratio be the ratio of a greater to a less, that is, let the side AB adjacent to the given angle be

greater than AC; and as in the last

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triangle ABC is given in species.

b 43. dat.

But if, in this last case, the given ratio of BA to AC be not the same with the ratio of ED to DG, that is, with the ratio of BA to the perpendicular AM drawn from A to BC; the ratio of BA to AC must be less than the ratio of BA to AM, 0 8. 5.

because AC is greater than AM. As BA to AC, so make

ED to DH; therefore the ratio of ED

to DH is less than the ratio of (BA to

A

AM, that is, than the ratio of) ED to DG; and consequently DH is greater P than DG; and because BA is greater than AC, ED is greater than DH. From the centre D, at the distance DH, describe the circle KHF, which necessarily meets the straight line EF in two points, because DHis greater than DG, and less than DE. Let the circle meet EF in the points F, K which are given, as was shown in the preceding case;

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and DF, DK being joined, the triangles DEF, DEK are given in species, as was there shown. From the centre A, at the distance AC, describe a circle meeting BC again in L: And

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