1000000030

G F.

]VI

t .\'o Ic.¡s in the oll,t"!'.

(,iz.

DE dOn OF. eaeh

ea~h

I dp~,q,\'dl'.

,..

OE. and

.'\e

OF ; and if Ihe

~ngles I n:!lId~d

b<t\\cen Ihe equdl legs

equdl.

VI :.

~:,e an~le

¡BC cqu.11

Ihe ,ogle

EDF ;

Ihen Ihe

r~nlllning.

leg of ILe olle

n1."1

be equal tOthe renuio–

lOg

I~g

Ibe olber.

t·i: .

El'; and Ihe "nglcs

<'pr clllc

cGG, lleg'

nt.I!

be equal.

vi! .

ABC equallo

DEF

(beiog orpolitc

the equ¡llegs AC and Or) . al;o

ACB equallo OFE (\\'Iu.:h are 0rpulile to Ihecqu.'¡ legs

AB and DE ). Por ir the triangle ABC be

fuppof~d

!tfl~d

up 2nd put upon Ihe Iriangle

DEF,

and Ihepoint

on Ihe point O ; il is

"in. r,nce

and DE are of

equallenglh, Ihe point E \ViII fal! upon Ihe point B; and

'lioee the aogb BAC EDF are equal. Ihe lille AC wifl

.f.1!

upon Ihe line DF ; and Ihey being of equall,nglh,

Ihe poinl C lI'il! fal! upoo the poinl

.nd fo the line HC

,,,ill exa81y agree lI'ilh the

lin~

EF. and the triangleABC

wil! inall refpe<'ls

exaéllyequal tO the triangle OEF;

and the angle ABC will be equallo the aogle DEI', alfo

1he angle ACB \Vil! be equal to Ihe angle DFE .

C,r.

l .

After Ihef2me manner il may be prored, Ihat

jf inany two triangles ABC, DEF. (fee the preceding

iigure) two angles

ABC

and ACB of Ihe one, be equ.1

to N'Oangles DEFand EFE or Iheolher. eaeh

eaeh

refpe8ively,

viz.

the angle ABC

the angle DEF, and

the 2ngle ACB equal to the angle DrE. and Ihe lides

iocluded belwcen Ihefe angles be .Ifo rqual•

•

iz,

BC e–

(jual

EF, then Ihe renldining aogles and Ihe r,des op–

por,te tO the equal angles, will alfo be equal eaeh tOeaeh

refpeélively;

viz.

Ihe aoele RAC equal tO the angle

.BDF. the r,de AB equal to OE, and AC equa] to

Df :

for if the triangle ARC be fuppofed tO be lirted up and

Iaid upon the triaogle DEF. Ihe poiol

bcing pUl upon

t he poiot

aod the line BC upon the line EF. r,neeBC

cnd EF

are

of equallenglbs, the point C \\ il! fall upon

Ihepoint F, and r,Dee the anele ACB is equal tOIhe 2n–

gle DFE, the line .CA will fall upon the line FD. aod

by the fame way of reafor.ing the line BA \ViII rallup-

the line EO ; and thererore the poiol of inl er–

fe8ion of the IWOlioes BA and CA.

viz.

A, will fdll

cpon the point of inlerfeélion of the IWO lines BO and

fO,

vito

D, and eonfequently BA will be equal tO DE.

~nd

AC equal ·to Dr, ano the

an~le

BAC tqual to the

angle EDF.

Coro

follO\\s likewife {rom this anicle, that if

.1ny triangle ABC (fig.

42 .)

has IWO of its r,des AB and

AC equal tOone anolher, the

an~les

oppoGte

Ihefe

lides

11',11

alfo

equal,

viz.

Ihe anglcs ABC equal to Ihe

lngle ACB. For fuppofe Ihe line AD. bifeéliog the angle

BAC, Or dividine it ioto I\VO egualancles BA)) and CAD,

and meeting BC in D,l hen

t!,~line

will divineIhe \Vhole

triangle IMC into t\VotrianglesABD andDAC; in whieh

CA and AD twO r,des of Iheone. ale equal tO CA and

AD two r,dcs of Ihe olher. eaeh tO eaeh refpe8ivcly.

l nd Ihe includcd angb BAD and DAC are hy fuppo–

(¡Iion equal; thmforc (I>y this afucie) Ihe angle ABC

muf! be c'luallo the

~ngle

ACH .

62 .An)

an.~/(,

HAD,(fig.

43 ')

all~!rireulII(mncr

cirel,

IMDE.

il bUI

.al[

Ihe alll:"

BC D

al Ih,

,,,,Ir(

j!andllll'.

Ih(

fÚII/(

a"'h

llED. To dllllonl¡'"e

",js,

d,aw thro'Jeh A aod u:e centre C. Ihe ri.nht

line ACE, the, the ,n.:le ECD is eqoal to bo:'h tbe

an~les

O.\C and ADG (by an. 59.) ; but r,nee AC and

CI)

are

equal (bci ng IWO radii of the ¡';me eirele) the

angles fubl cn.l!J by them muf! be eq ual alfo.

(byarl,

62. coro

2.)

Ihe

an~le

C.'\O equ.d to the angle

l'D.-\;

thedure Ihe fum of th.m is double aoy one of

Ihem.

D:\C 2nd ADC is double

CAD. and

therefore ECO is

alto

double of

DAC :

Ihe f,me way

it may be proven, thal ECB is double of CAB; and

Iherefore Ihe an¡;le BCD is double of Ihe anele

HAO,

Or BAO the half of BCD, whieh was

be proved.

C,r.

l .

Heoee an angle al rbe eireumferenee is mea–

fured by half the are it fubtends; for the angle at Ihe

centre ('bnding on the faOle are) is mea fu red by Ihe

whole are(by

art.

29 ) ;

bUI finee the angle al theeenlre

is double Ihat at Ihe eireumferenee. ir is plain Ihe .ngle

the eircumfcreoee mufi be me2[ured by only half Ihe

8re il fi.nds UpOD.

Coro

2 .

Henee all angles. ACB.

DB. AEB.

&c.

(fig.

44')

at Ihe c.ireumfcrenee of a cirele. I!andineon rhe

lame ehord AB, are equal

one aoother; for by the

Idlf eoroilary rhe)' are aU meafured by Ihe fame are.

viz,

half the are AB whieh eaehof thrm fubtends.

C.r.

Henee an_ngleinafegmentgreaterthan afemieir.

cleis lefsIhanarigh!20gle: Ihus,ifADB

afegmenl great–

enhan afemieirele, (feeIhelal! figure) thanrhe are AB. on

whieh it fi ands, mulf be lefs thaDa femieirele. and Ihe

h.lf

of it lefs rhan a quadrant or a right angle; but Ihe

angle ADB io the {egment is meafured by the half

AH. therefore it is lefs than a right angle.

Coro

An angle ina femicirele is a right angle, For

r,nee ABD (fig.

45 .)

is a femieircle, the are AED mu(l

alfo be a femieircle: but the angle ABD is mea fu red by

half the are AED. that is, by half a femieircle or qua–

dranr; Iherefore the angle ABD is a right one.

Coro

Henee an aogle in a fegment lefs Ihan a femi–

eirGle. as ABO. (fig.

46.)

is greater thana right anglc-:

for r,nee the are ABD is lefs thao a femieircle. the are

ED mufi be greater Ihao a femieirde. and fo il is half

grraler Ihan a quadraot.

e.•

rhan

tllC

me. fure of a right

angle; therefore Ihe aogle ABO. II'hieh is meafured by

halr Ihe are AED. is grcater than aright angle.

63 ./ffromlh( cmtr(Cofth(circleABE,(6g. 47 ·) lhm

/<1

f." Ihr p,rptndicu/dr

on Ih( chord

AB,

Ihm

Ih,t ""pendicular wi" bijdl th( chord

in Ih, poinl

D.To

demonllrate Ihis. draw from the centre

Ihe

extremilics of Ihe ehord the t\Vo lines

CA,

CH ; Ihco

r,nee ,he lines CA and CB are egual. thr angles CAH.

CHA•

hich Ihey fubtend mufi be equal alfo,

(byarl.

6~ .

coro

2.)

but Ihe perpendicular CD divides Ihe trian–

gle ACH iototlVO right-aogled Iriangles ACO and CDB.

in \Vlueh the fum or the angles ACD and

CA D

in Ihe

one, isequal

rhefum of Ihe angb OCB and CIl Din

Ihe olher. eaeh

bcin~

equal

a righl

a~gle,

(by

coro

01'1.

6\.)

but CAD is equal to CIl O. Iherefore ACD

c~ual

!lCD So in the t\\lO tri"ngb

ACD

and

nCD. the I\VO legs i\C and CD in Ihe one, ate equallo

Ihc I\VO legs

. nn C' D io Ihe olher. eaeh

caeh re–

fpe8ivcly, ;md Ihe includtJ

.ngl~s ,~C'D

and BCD are

lq ual; Ihcrdurc Ihe r,m"ining 1"1;s AD ooJ HD Me

c~u,tI

(by ort. GJ ) anJ

cor.fe'

l~·:otly

.'\B

bir<8~d

in D.

('4'