ven, wherefore each of thofe at the points D, E, F is given: And because the straight line FD is drawn to the given point Din DE which is given in pofition, making the given angle EDF; therefore DF is given in pofition. In like manner EF b 32. dat. alfo is given in pofition; 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 triangle e 29. dat. DEF is given in fpecies; and it is fimilar to the triangle d 42 dat. ABC; which therefore is given in fpecies.

PROP. XLIV.

I one of the angles of a triangle be given, and if the fides 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 fides BA, AC about it have a given ratio to one another; the triangle ABC is given in fpecies.

Take a ftraight line DE given in pofition and magnitude, and at the point D in the given ftraight line DE, make the angle EDF equal to the given angle BAC; wherefore the angle EDF is given; and becaufe the ftraight line FD is drawn to the given point D in ED which is given in position, making the given angle EDF; therefore FD

And because

is given in pofition.
the ratio of BA to AC is given,
make the ratio of ED to DF the

fame with it, and join EF; and be-
cause the ratio of ED to DF is gi- B

4. 6.

1. def.

41.

6.

A

ven, and ED is given, therefore b D is given in magnitude; b 2. dat. and it is given alto in pofit on, and the point D is given, wherefore the point F is given; and the points D, E are given, c 30. dat. wherefore DE, EF, FD are given in magnitude; and the d 29. dat. triangle DEF is therefore given in fpecies; and because the e 42. dat. triangles ABC, DEF have one angle BAC equal to one angle EDF, and the fides about these angles proportionals; the triangles are f fimilar; but the triangle DEF is given in fpecies, f 6. 6. and therefore alfo the triangle ABC.

See N.

42.

a 2. dat.

22. 5.

C 20. I.

d A. 5.

e 22. I.

IF

PROP. XLV.

F the fides of a triangle have to one another given ratios; the triangle is given in fpecies.

Let the fides of the triangle ABC have given ratios to one another, the triangle ABC is given in fpecies.

Take a ftraight line D given in magnitude; and because the ratio of AB to BC is given, make the ratio of D to E the fame with it; and D is given, therefore E is given. And because the ratio of BC to CA is given, to this make the ratio of E to F the fame; and E is given, and therefore F; and because as AB to BC, fo is D to E; by composition AB and BC together are to BC, as D and E to E; but as BC to CA, fo is E to F; therefore, ex aequali, as AB and BC are to CA, fo are D and E to F, and AB and BC are greater than CA; therefore D and E are greater than F. In the fame manner any two of the three D, E, F are greater than the third.

Make the triangle GHK whose

A

BL

10

C

Ꮐ

DEF

fides are equal to D, E, F, fo that GH be equal to D, HK tồ E, and KG to F; and becaufe 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 fpecies: But as AB to B, fo is (D to E, that is) GH to HK; and as BC to CA, fo is (E to F, that is) HK to KG; therefore, ex aequali, as AB to AC, fo is GH to GK. Wherefore the triangle ABC is equiangular and fimilar to the triangle GHK; and the triangle GHK is given in fpecies; therefore alfo the triangle ABC is given in fpecies.

85. 6.

COR. If a triangle is required to be made, the fides of which fhall have the fame ratios which three given ftraight lines D, E, F have to one another; it is neceffary that every two of them be greater than the third.

PROP.

PROP. XLVI.

IF the fides of a right angled triangle about one of the acute angles have a given ratio to one another; the triangle is given in fpecies.

Let the fides 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 fpecies.

43.

Take a straight line DE given in pofition and magnitude; and because the ratio of AB to BC is given, make as AB to BC, fo DE to EF; and because DE has a given ratio to EF, and DE is given, therefore EF is given; and because as AB a 2. dat. to BC, fo is DE to EF; and AB is lefs than BC, therefore DE is less than EF. From the point D draw DG at right angles c A. 5. to DE, and from the centre E at the diflance EF, defcribe a circle which fhall meet DG in

с

19. I.

A

D

two points; let G be either of them, and join EG; there

F

fore the circumference of the

circle is given in pofition;

and the straight line DG is given in pofition, because it ise 32. dat. drawn to the given point D in DE given in pofition, in a given

angle; therefore f the point G is given; and the points D, E f 28. dat. are given, wherefore DE, EG, GD are given in magnitude, g 29. dat. and the triangle DEG in fpecies h. And becaufe the triangles h 42. dat. ABC, DEG have the angle BAC equal to the angle EDG, and the fides about the angles ABC, DEG proportionals, and each of the other angles BCA, EGD lefs than a right angle; the triangle ABC is equiangular and fimilar to the triangle i 7. 6. DEG: But DEG is given in fpecies; therefore the triangle ABC is given in fpecies: And in the fame manner, the triangle made by drawing a straight line from E to the other point in which the circle meets DG is given in fpecies.

PROP.

Sce N.

44.

PROP. XLVII.

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

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

First, Let the given ratio be the ratio of equality, that is, let the fides BA, AC, and confequently the angles ABC, ACB, be equal; and because the angle ABC is given, the angle ACB, and alfo the remaining angle BAC is given; therefore the triangle ABC is b 43. dat. given in fpecies: And it is evident that in B this cafe the given angle ABC must be acute.

a 32. I.

A

A

C

Next, Let the given ratio be the ratio of a lefs to a greater, that is, let the fide AB adjacent to the given angle be lefs than the fide AC: Také a ftraight line DE given in pofition and magnitude, and make the angle DEF equal to the given c 32. dat. angle ABC; therefore EF is given in pofition; and because the ratio of BA to AC is given, as BA to AC, fo make ED to DG; and because the ratio of ED to DG is given, and ED is given, the ftraight line DG is gi ven, and BA is lefs than AC, therefore ED is lefs than DG. From the centre D, at the distance DG defcribe the circle GF meeting EF in F, and join DF; and because the circle is given f in pofition, as also the straight line EF, 28. dat. the point F is given; and the points D, E are given, wherefore

d 2. dat.

e A. 5.

f 6. def.

e

the ftraight lines DE, EF, FD

E

B

D

F

h 29. dat. are given in magnitude, and the triangle DEF in fpe42. dat. cies, and because BA is lefs than AC, the angle ACB is k 18. I. lefs than the angle ABC, and therefore ACB is lefs than

II. 7. I.

a

a right angle. In the fame manner, because ED is less than DG or DF, the angle DFE is lefs than a right angle: And because the triangles ABC, DEF have the angle ABC equal to the angle DEF, and the fides about the angles BAC, EDF proportionals, and each of the other angles ACB, DFE lefs than a right angle, the triangles ABC, DEF are fimilar, and m 7. 6. DEF is given in fpecies, wherefore the triangle ABC is also given in fpecies.

B

A

C

c 32. dat.

Thirdly, Let the given ratio be the ratio of a greater to a lefs, that is, let the fide AB adjacent to the given angle be greater than AC; and, as in the laft cafe, take a ftraight line DE given in pofition and magnitude, and make the angle DEF equal to the given angle ABC; therefore EF is given in pofition: Alfo draw DG perpendicular to EF; therefore if the ratio of BA to AC be the fame with the ratio of ED to the perpendicular DG, the triangles ABC, DEG are fimilar ", because the angles ABC, DEG are equal, and DGE is a right angle: Therefore the angle E ACB is a right angle, and the triangle ABC is given in fpecies.

A

G F

b 43. dat.

But if, in this laft cafe, the given ratio of BA to AC be not the fame 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 muft be lefs than the ratio of BA • 8. 5. to AM, because AC is greater than AM. Make as BA to AC fo ED to DH; therefore the ratio of ED to DH is lefs than the ratio of (BA to AM, that is, than the ratio of) ED to DG; and confequently, DH is greater P than DG; and becaufe BA is great. B er than AC, ED is greater than DH. From the centre D, at the distance DH, defcribe the circle KHF which neceffarily meets the ftraight line EF in two points, because DH is greater than DG,

e

and lefs than DE. Let the circle meet E K

EF in the points F, K which are given,

C

P 10. 5. e A. 5.

as was shown in the preceding cafe; and, DF, DK being joined, the triangles DEF, DEK are given in fpecies, as was the

Cc