ments 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 a; and CD is given a 33. dat. in position, wherefore the point M is given b; and the point Lb 28. dat. is given, LM is therefore given in magnitude; and because c 29. dat. the ratio of GE to GF is given, and as GE to GF, so is NL to

NM; the ratio of NL to NM is given; and therefore the ratio of ML to LN is given; but LM is given in magnituded, wherefore LN is given in magnitude; and it is also given in position, and the point L is given; wherefore f the point N is f given; and because the straight line HK is drawn through the given point N parallel to CD which is given in position, therefore HK is given in positions.

PROP. XLI.

Cor.

6. or

(7.dat.

e 2. dat.

30. dat.

g 31. dat.

F.

IF a straight line meets three parallel straight lines see N. which are given in position, the segments into which they cut it have a given ratio.

Let the parallel straight lines AB, CD, EF, given in position, be cut by the straight line GHK; the ratio of GH to HK is given.

c 29. dat. fore AH is ge angle

ratio of
fore the

E

because FG to the angle BAC; therefore, because the anto AD, sich is equal to the angle BAC, has been found, BAC is givenc; in like manner the angles at B, C And because the sides AB, BC, CA are given, alios to one another are given d, therefore the triangle e 30. dat. give given e in species.

d 2. dat. nitud

f 31. dat.

、

23. 1.

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.

Take a straight line DE given in position and magnitude, and at the points D, E make a the angle EDF equal to the angle BAC, and the angle DEF equal to ABC; there- B fore the other angles EFD, BCA are

A

D

CEF

equal, and each of the angles at the points A, B, C, is 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 EF also b 32. dat. 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 triangle c 29. dat. DEF is given in species; and it is similare to the triangle d 42. dat. ABC: which is therefore given in species.

4. 6. e1. def. (6.

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.

A

D a 32. dat.

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 position, making the given angle EDF; therefore FD is given in position. 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 because the ratio of ED to DF is given, B and ED is given, therefore b DF is given in magnitude: and it b 2. dat. is given also in position, and the point D is given, wherefore the point F is given; and the points D, E are given, where- c 30. dat. fore DE, EF, FD are given din magnitude; and the triangle d 29. dat. DEF is therefore given e in species; and because the triangles e 42. dat. ABC, DEF have one angle BAC equal to one angle EDF, and the sides about these angles proportionals; the triangles are f similar; but the triangle DEF is given in species, and f 6. 6. · therefore also the triangle ABC.

C E

F

!

See N.

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 a 2. dat. the same; and E is given, and therefore a F; and because as AB to BC, so is D to E; by composition AB and BC together are to BC, as D and E to F;

c 20. 1. d A. 5.

but as BC to CA, so is E to F;

H K

A

D E F

b 22.5. therefore, ex æqualib, as AB
and BC are to CA, so are D
and E to F, and AB and BC
are greater than CA; there- B
fore D and E are greater than
F. In the same manner any
two of the three D, E, F are
e 22. 1. greater than the third. Make e
the triangle GHK whose sides
are equal to D, E, F, so that GH be equal to D, HK to E, and
KG to F; and because D, E, F are each of them given, there-
fore GH, HK, KG are each of them given in magnitude;
£ 42. dat. therefore the triangle GHK is given fin 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 aquali, as AB
to AC, so is GH to GK. Wherefore & the triangle ABC is
equiangular and similar to the triangle GHK; and the trian-
gle GHK is given in species; therefore also the triangle ABC
is given in species.

g 5. 6.

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.

1

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.

Take a straight line DE given in position and magnitude; - and because the ratio of AB to BC is given, make as AB to BC, so DE to EF; and because DE has a given ratio to EF, and DE is given, thereforea 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 EF, describe a circle which shall meet DG in two points; let G be either of them, and join EG; therefore B

A

F

C

straight line DG is given in position, because it is drawn to e 32, dat. the given point D in DE given in position, in a given angle; therefore the point G is given; and the points D, E are given: f 28. dat. wherefore DE, EG, GD are given & in magnitude, and the tri- g 29. dat. angle DEG in speciesh. And because the triangles ABC, DEG ĥ 42. dat. have the 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 equiangulari and similar to the triangle DEG: but i 7. 6. DEG 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.