From the given point A let the straight line AED be drawn to the two parallel ftraight lines FG, BC, and let the ratio of the fegments AE, AD be given; if one of the parallels BC be given in pofition, the other FG is alfo given in pofition.

From the point A draw AH perpendicular to BC, and let it meet FG in K. and becaufe AH is drawn from the given point A to the ftraight line BC given in pofition, and makes a given angle

A

B

FE

K G

a. 33. Dat. AHD; AH is given in pofition. and BC is likewife given in pofition, thereb. 28. Dat. fore the point H is given. the point A is alfo given, wherefore AH is given in e. 29. Dat. magnitude . and, becaufe FG, BC are parallels, as AE to AD, fo is AK to AH; and the ratio of AE to AD is given, wherefore the ratio of AK to AH is given; but AH is given in magnitude, therefore AK is given in magnitude; and it is ano e. 30. Dat. given in pofition, and the point A is given; wherefore the point K is given. and becaufe the ftraight line FG is drawn thro' the f. 31. Dat. given point K parallel to BC which is given in pofition, therefore FG is given in pofition.

d. 2. Dat.

37-38.

See N.

PROP. XL.

F the ratio of the fegments of a ftraight line into which it is cut by three parallel ftraight lines, be given; If two of the parallels are given in pofition, the third alfo is given in pofition.

Let AB, CD, HK' be three parallel ftraight lines, of which AB, CD are given in pofition; and let the ratio of the fegments GE,

GF into which the ftraight line GEF is cut by the three parallels, be given; the third parallel HK is given in pofition.

In AB take a given point L, and draw LM perpendicular to CD, meeting HK in N. becaufe LM is drawn from the given point L to CD which is givea in pofition, and makes a given angle IMD; LM is given in pofition 2. and CD is given in position, wherefore the point M is given b; and the point L is given, LM is therefore given in magnitude . and becaufe the ratio of GE to

2. 33. Dat.

b. 28. Dat.

c. 29. Dat.

d.

or

7.Dat.

GF is given, and as GE to GF, fo is NL to NM; the ratio of NL to NM is given; and therefore the ratio of ML to LN is Cor. 6. given. but LM is given in magnitude, wherefore LN is given in magnitude; and it is alfo given in pofition, and the point I is given; wherefore f the point N is given. and because the ftraight f. 30. Dat. line HK is drawn thro' the given point N parallel to CD which is given in pofition, therefore HK is given in pofition ".

PROP. XLI.

e 2. Dat.

8. 31. Dat.

F.

IF a ftraight line meets three parallel ftraight lines See N. which are given in pofition; the fegments into

which they cut it, have a given ratio.

Let the parallel ftraight lines AB, CD, EF given in pofition be cut by the ftraight line GHK; the ratio of CH to HK is given. In AB take a given point L, and draw LM perpendicular to CD, meeting EF in N; therefore LM is given in pofition; and CD, EF are given in pofition, where- C

G I

fore the points M, N are given. and the
point L is given, therefore the ftraight,
lines LM, MN are given in magnitude; E K
and the ratio of LM to MN is therefore

a. 33. Bat.

b. 29. Dat.

N

e. 1. Dat. given. but as LM to MN, fo is GH to HK; wherefore the ratio of GH to HK is given.

39.

See N.

2. 22. I.

IF

PROP. XLII.

each of the fides of a triangle be given in magnitude; the triangle is given in species.

Let each of the fides of the triangle ABC be given in magnitude; the triangle ABC is given in fpecies.

Make a triangle DEF the fides of which are equal, each to each, to the given ftraight lines AB, BC, CA; which can be done, because any two of them must be greater than the third; and let

B

A

CE

DE be equal to AB, EF to BC, and FD to CA. and because the two fides ED, DF are equal to the two BA, AC, each to each, and the bafe EF equal to the bafe BC; the angle EDF is equal to the angle BAC. therefore because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is given, in like manner the angles at B, C are given. and because the fides AB, BC, CA are given, their ratios to one another are given. . 3. Def. therefore the triangle ABC is given in fpecies.

b. 8. 1.

c. 1. Def.

d. 1. Dat.

a. 23. 1.

IF

F each of the angles of a triangle be given in magnitude; the triangle is given in fpecies.

Let each of the angles of the triangle ABC be given in magnitude; the triangle ABC is given in fpecies.

Take a ftraight line DE given in pofition and magnitude, and at the points D, E make the angle EDF equal to the angle BAC, and the angle DEF equal to ABC; therefore the other angles EFD, BCA are equal.

B

A

D

CEF

and each of the angles at the points A, B, C is given, wherefore

each of thofe 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 pofition, making the given angle EDF; therefore DF is given in pofition b. in like manner EF alfo is given in pofition; wherefore b. 32. Dat. the point Fis given. and the points D, E are given; therefore each

c

of the ftraight lines DE, EF, FD is given in magnitude. where- c. 19. Dat. fore the triangle DEF is given in fpecies; and it is fimilar to the d. 42. Dat. triangle ABC; which therefore is given in fpecies.

PROP. XLIV.

IF 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 fpecies.

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.

A

D

4. 6.

1. Def. 6.

41.

a. 32. Dat.

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 because the straight line FD is drawn to the given point Din ED which is given in pofition, making the given angle EDF; therefore FD is given in pofition .. and becaufe the ratio of BA to AC is given, make the ratio of ED to DF the fame with it, and join EF. and because the ratio of ED to DF is given, and ED is given, therefore DF is given in magnitude; and it is given alfo in pofition, and the point D is given, wherefore the point F is given . and the points D, E are given, e. 30. Dat. wherefore DE, EF, FD are given in magnitude; and the triangle d. 29. Dat. DEF is therefore given in fpecies. and because the triangles ABC, c. 42. Dat. DEF have one angle BAC equal to one angle EDF, and the fides

B

CE Fb. 2. Dat.

about these angles proportionals; the triangles are f fimilar. but f. 6. 6, the triangle DEF is given in fpecies, and therefore alfo the triangle ABC.

Ir

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.

2

Take a ftraight line D given in magnitude; and because the ratio of AB to LC 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 fime; and E is given, and therefore F. and becaufe as AB to BC, fo is D to E, by compofition AB and DC together are to BC, as Dard E to E; but as BC to CA, E to F, therefore, ex aequali, a, AB and BC are to CA, foare D and E to F. and AB B and BC are greater than CA, then fo e Ded E are greater than F. in de fine nacr any two of the thee D, E, F are greter than the chind. take © the

J

G DEF

K

tinngle Giak who's fides are coud to D, E, F, fo that GH be equal to D, HK to E, and KG to F. and becaufe D, E, F are, each of them, given, therefore GH, HK, KG are each of them f. 41. Dat given in magnitude; therefore the triangle GHK is given fin fpecies. but as AB to BC, 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 acquali, 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.

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.