foning, it may be fhewn that AB AD :: AC: AE. Q. E. D.

e

d 185.

The second Part of the Theorem may be thus demonftrated. By the Suppofition, as BD: DA :: CE EA; and as BD: DA :: A BDE: A 189. ADE; and as CE EA :: ACDE: AADE; :. As ABDE: ADE :: ACDE: AADE... the As BDE, CDE, have the fame Ratio to the A ADE, and the ABDE ACDE, and stands 181. on the fame Base DE: But it is manifest that equal As muft have equal Areas, and as the Area of a A is its Bafe into its Altitude, and they have the fame Base, they must consequently have the fame Altitude, otherwise their Areas could not be equal; that is, the L Diftance of the Points D and E, from the Line BC, are equal, and... DE to || BE. Q. E. D.

f

192. THEOREM 5. If the Angle BAC, of a Triangle ABC, be divided into two equal Angles by a Line AD, cutting the Bafe, the Segments of the Bafe will bave the fame Ratio that the other two Sides of the Iriangle have to one another; viz. as BD: DC :: AB: AC. And if the Segments of the Base bave the fame Ratio which the other Sides of the Triangle have to one another, the Line AD, drawn from the Vertex to the Point of cutting, will divide the BAC of the Triangle into two equal Parts; viz. LBAD= LCAD.

I

ACE =

E: But

BAC = 4DAC,

First. Let BA be produced till AE = AC, and let CE be joined; then, the the BAC LACE + LE, and as these s are equal, each of them must be that is, DAC = LACE; to each other, . As BD : Q. E. D.

112.

$ 96.

DC d

AD, EC, are 73. :: BA: AE.

c

d

191.

C

Secondly. By the Suppofition, BD: DC :: AB
AC; but AC AE by the Conftruction; '.' as

BD: DC :: AB AE; AD is to EC;

:

* 40.

72. a 59.

45.

193. THEOREM 6. If four Lines, or Numbers, denoted by the Letters, a, b, c, d, are proportional, viz. as a b c d; the Rectangle contained by the Extremes is equal to the Rectangle under the Means, viz. a x d = bxc. And if the Rectangle contained by the Extremes be the Rectangle contained by the Means, thefe four Quantities are proportional; that is, if a x dbx c, then it will be, as a b c : d.*

c : d,

But bxc, and

... Firft. Since a x d, and b xd, are Equimultiples of a and b, we have, as a x d : b x d 2 : à : b. a 187. And fince by the Suppofition, a b 185. we have a xd: b x d ' :: c : d. bx d are Equimultiples of c and d, : bx c : b x dc :: c : d; . a × d : b x d :: 1 b x ċ : b x d ; • a x d = b xc. Q.E D.

c 187.

d 185.

* 181.

d

d

Secondly. It is plain, that a x d, b x d, are Ef 187. quimultiples of a and b, a ba x d : b x and b xc, bx d, are Equimultiples of c, and d; c d :: bx c : bx d; but a x d = b x c by the Suppofition, . by writing a xd, for its equal bc, we have, as c d :: a x d : b x d ; but it has been juft fhewn that a baxd: b× d, . c : d has the fame Ratio as a b, that is, as a b c d. Q. E. D.

5.194.

*This Theorem may be demonftrated algebraically thus. First,

gives dabe. Q. E. D. Secondly, If we divide each Side of

bc

the Equation da be by d, we have a = and this again by b

d'

194. THEOREM 7. Equiangular Triangles are fimilar; that is, the Sides about the equal Angles are Proportionals.

Let Da AB, Db AC, then the A by the Suppofition, the As Dab, equal in every Refpect;. Dab ZABCDEF by the Supposition,

D being
ABC, are *

a F.

PI. IV. 7.

2 58.

45.

4 ABC; but 2

·.· ▲ Dabb = 。

40.

DEF, ba to EF; DE: Da :: * DF : Db. ·. for Da, Db, writing their equals AB, AC, * 196. the Proportion becomes DE: AB :: DF : AC. Q. E. D..

And if Fd be taken CA, and Fc CB, it may be fhewn in the fame Manner, that FD : CA :: FE CB. Q. E. D.

By joining these two Proportions in one, we have DE: AB: DF AC: FE: CB. Q. E. D.

:

195. THEOREM 8. If the outward Angle CAE, of Pl. IV. a Triangle ABC, is bifected by a Line AD, cutting the F. 8. Bafe BC produced in D; then the Segments BD, DC, will have the fame Ratio as the other Sides of the Triangle; viz. BD : DC :: BA: AC. And if BD : DC :: BA: AC, the Line AD will bifect the outward Angle CAE, viz. ¿CAD = 2 DAE,

d

CFA2 = 2
but the L
CFA = 45.

to

For let CB be I to DA; then
DAE, and alfo LACF DAC;
DAE DAC by the Suppofition,
LACF, AF AC. Now, as CF is
AD by the Suppofition, it follows, BD: DC
BA AF; but it has been juft fhewn that AF =
AC, for AF writing its equal AC, the above
Proportion becomes BD DC: BA: AC.
Q. E. D.

Secondly. By the Suppofition, BD
AC, and FC being to AD, BD :

::

DC: BA:
DC :: BA :

AF, . BA AC: BA AF; AC AF,

c

72.

d 62.

72. DAE, and ACF CAD, ... ¿DAE (= €45. ACF) CAD. Q. E. D.

F. 9.

196. THEOREM 9. If two Triangles, ABC, DEF, Pl. IV. have one Angle A, of one, equal to one Angle D, of the other, and the Sides containing the equal Angles proportional, viz. AB AC :: DE: DF, the Triangles are equiangular and fimilar.

Let Af be Df, and Ae DE; and let fe bể joined. Then the LA being the D by the 58. Suppofition, the As Afe, DFE, are * equal in every Respect. By the Suppofition, AB AC :: DE: DF; or, which is the fame, by writing equals, (viz. Ae for DE, and Af for DF ;) AB : 191. AC Ae: Af. fe is to BC, Afec= 40. Aef=▲ABC; LACB, and ▲ Aef= LABC;

194.

Fl. IV.

·.· ▲ the As Afe, ABC, are equiangular, and confequently, as the As Afe, DFE, have been juft fhewn to be equal in every Respect, ABC, D'EF, are alfo equiangular and also fimilar. Q. E. D.

197. THEOREM IO.

1

If two Triangles, ABC, DEF, have their like Sides proportional, (thus, AB : F. 9. AC DE DF; and AB BC DE: FE.) the Triangles are fimilar to each other.

191.

b

Let Af DF, and Ae DE, and let fe be joined; then by the Suppofition, AB AC :: DE : DF; and by writing equals, viz. Ae for DE, and Af for DF, the Analogy is, AB : AC :: Ae: Af; fe is to BC. Again, by the Suppofition, AB BC :: DE : FE; and the As ABC, 196. Aef. being equiangular and fimilar, AB : BC :: Ae fe; by Equality of Ratios, DE FE:: Ae fe; but Ae DE by the Conftruction, .. writing DE for its equal Ae, we have, DE FE :: DE fe, confequently FE fe. Hence the three =fe. Sides of the As Afe, DFE, have been proved to be refpectively

c 185.

refpectively equal to each other, and

d

the As Afe,

DFE, are equal in all Refpects; and confequently,
as the A Afe is fimilar to the AABC, the ADEF
muft be fo too.
Q. E. D.

198. THEOREM II. If four Quantities are proportional, they will also be Proportionals when taken inverfely; viz. if a b c d, then alfo, b: a :: d: c.

For the Rectangle of the Extremes being the Rectangle of the Means, a x d = bx c ; or, which is the fame, bx c = ax d, ·.. b : a * :: d : c. xd, Q. E. D.

199. THEOREM 12. If four Quantities be proportional, they will also be proportional when taken alternately; that is, if a b c d, then a c :: b : d.

For the Rectangle of the Extremes being * Rectangle of the Means, a x dbx c, or, which is the fame, a x d = c xb, ... a : c :: b: d. Q. E. D.

**

:

200. THEOREM 13. If four Lines, denoted by a, b, c, d, are Proportionals, viz. a b c d; they will also be compoundedly proportional; that is, a + b : 5 :: c+d: d.†

Let AB = a, BC b, AE = c, ED=d; Pl. IV. then is AC = a + b, and AD=c+d. Because Fig. 10.

Algebraically demonftrated thus. For the Product of the Extremes, a + bxd = ad + bd, and the Product of the Means, b x c+d= bc + ba; but a b: cd, by the Suppofition; ... ad = bc, confequently ad+bd= bc + bd, or, which is the fame, a+b x d = b xc+d, and . by Art. 193. a+b : c+d: d. Q. E. D.