689 / 1042
le
o
~ ; .
li f,,)o) ,he c. ntlc C "f a ei,el.:
.~
I;E, Ihere
~e
.¿r." o
a
p" r,·nJ.eul.,
<.
O
00 the ehord ,\Il, aoJ
"ro.luc«1
"JI
i, me<!1
,h~
ci,de in
F.
,hell ,h< lioe
CF
b'¡~ds
,he ."eh Al! tO ,h< puinl F; (oor (1;'<,he (ore·
going I;gu,e)
joiuin.~
,lte poiots :\. auJ
F.
Jo'
. 0J
11
by Ihe
11,(;~llt
I:oes
,\1',
I'H,
theo io ,he triJ"gles .-\DF,
IJOi",
:\.D is <9".1110 DB (by
u' .
63.)
aod DI' cummoo
tO
borh; ,he,dore AD ar.d DI", {I,O
le~s
oi' ,he triaoglo
ADr. are ,qll.1 lO
RO
~od
Dr'.
,wn
Ie;;s of ,he '''.10'
gle BDf. ami ,he iocludeJ angl<s ADt' BD1' are cqudl,
bc:og hnth ,ighl; rherdore (hy arto6 I ) Ihe remainiog
legs
AF
and
fll
are equ?l; bu, in ,he (lIl1e e'rele equd l
lio~s
are ehords of equal arches, ,hercfor<! Ihe arches
AF
and
Fll
are equ31.
So ,he
\l'h~le
3rch
11
t'[j
is bilctleJ
io ,he point F by ,he lioe CF.
e".!.
From
.rt.
6J. i, fullol"s,
Ih.tIany li"e bifetl
ing a ehord at righ l aogles is a
dlam~t<r ;
for (,oce
.(hy
atto
6] .)
a lioe'dralV from Ihe eeotr< perpendICular to
a
chord, bifetls thal chord
31
righ, a0
6
Ie,; ,hec<!!ore, eoo·
"erO)', a Jine bi(etling a ehord at righ, aoglts. mul!
p.,!s
thro' ,he ceotre, aod eoofequeotly be a didmeter
eor.
2.
From Ihe two laU anides i, follow,.
t:'~t
,he (,ne
of aoy are is the half of Ihe eho,.1 uf twie<
,h~
, re; for
((ee Ihe foregoiog (,heme) AD is the (,ne of the are
A
1",
by the
d~(;nition
of a
fin~,
and Al' is half ,he are
AFB, aod AD half the ehord AB (by art.
63 ') ;
thm·
fore ¡he
coro
is
pl.in.65 . In 2n.y
tri3ng!~,
the half of each fide is Ihe fin e
of the oppo(,te angle; for if a circle be Cuppofed to be
dralVo through ¡he ¡hree aOEul. r roinl!
!l,
R,
and
D
of
the triangle ABO,
6;;. 48.
Iheo the
an~le
DAB is mea·
fured by half ,he areh BKD (by
cer.
l. oí
arl. 62 .)
hut ¡he half of BD,
viz.
BE,
is ¡he (,ne of
h.df,he .reh
BKD.
vÍl
¡he (,ne of
BK
(by
roro
2 .
of ¡he lall) whieh
i, ,he meafure of the aogle' BAD; theref"re ¡he half of
BD is the fine of ¡he angle B." D
¡
the fa me way it may
be proved, that ,he half of A
D
is ,he fioe of the angle
ABD, and the half of
A
B is ,he fine of th< angle
"DB.
66.
The
(,n~,
tangent.
&c.
of aoy areh is called alfo
the fine, tangenl,
&c.
of the angle whofe mea(u re the
areis: thus
beca~fe
theareGD
(fig' 49')
is ,he meafure
of ¡he angle GCD; and fince GH is ,he (,oe, DE the
ungent, HD ¡he verfed (,ne. CE the (ecaot, al(o GK
the eo·fine, BF the eo langeot. and CF Ihe eo·fecan¡,
Oc.
of Ihe areh GD; ¡hen GH is colled ¡he (,ne, DE
the tangen,.
Cc.
of ,he anglc GCD, who(e lDeafure is
the mh GD.
67.IfI'um qlla!
ond
tara!'"!inu.
AB
,lndCD
(fig'50.)
b,
j ,intd h¡ l'UJa olhm,
AC
nnd
liD;
Ih", Ihc(t)/¡J/I
al·
fob<
'qua!and/,aralld.
To demonflrate tllIs, jo:o Ihe,wo
0pp0(,Ie aoglcs
/1
,nd
J)
wi,h Ihe line
A
O;
¡hen i¡ ;s
pldin Ihis lioe :\D divides Ihe
~ "3Jrdmral .
ACD8. into
two
tri~ngles,
viz.
AH D, ACD, in wllleh A!l a leg of
the onr, isequal tODC a
le~
01' ¡he oth!r, hy ("ppofit;.
00, aod AD i, eO'"01on tO bnlh tri., 1:lcs; , ud fi"ce :\
H
i, pmlld
ro
(, D,
the angle
B .~
1)
,"11
be 'qual
lO
tite
aogle A
DC,
(ily .rt. ,/,.) ¡hercfure in Ihe tWO
tri.ln·
~I.,
HA
aod AD : allo! ¡he , nglc
H:\ D
is e(l"al
to Cl) ano!
Di\.;
and Ihe
u.~!e
ADC, Ihal i.,
111'0
le~s
ano! Ihe inrl l1,kJ angle io Ihe OIlP, i,
"9" 01
tll
'"0
Jeg,: :tnd 1I1t.'
;ndut!cd
~ nglc
in
lile orhl r; th"l l'(' 'I'"
(1
y
..l!.
61 )
POO
is cqll.d
1"
AC, .nd fi "ee tlt.
"n~k
D '\(
VOL .
lf. No.
5 ~ .
E
'1'
I~
Y.
is cqual to ,heanglc .'\D11.
Ih~refvre
¡1,e :;ocs
TI;) ¡\C
:m
pJr. lld (by
,·or. arl .
59' )
. e,r.
t. Heoee jt is pLI:n. lha, Ihe qt:a¿,i:d'erJl
ABOC
IS
a
par~lldogr2r.l,
(,uC<
Ihe 0pp0(,le lidcs a,e p. r.d!d.
e.r.
2.
In any PM.lld ogrólm
Ih~ ¡'n~
j
.io,nc Ihe
o,.'
po(,te angb (eallcJ ¡he diJgon.
.!)
di
AD, u"':des t!,e
fi.
gure iOlo ¡WOequal pans, lince i¡ 1m been prol'd ,he!
Ihe ¡riaogles ABI) AC D are equal¡o one , "olher.
e.r.
3.
J¡
follo", allo, ,ha, a triaogle Ae D un ,he
(ame ba(e CD, an.! b=Iween the (Jl1Ie par. lld,
\\ :ú
a
paralldogram ABDC, is the h•
.! f
uf th.¡ pm lklo ·
¡:ram.
Coro
4.
Henee j, is plain, ,hat the oppo(,te (¡des of a
paralldogram are equal ; fvr il h,s
be~n
profld.
'~H
/IB DC bciog a
p3rallel~~ram,
AB w.1I be equll to CD,
and AC equ<1
10
BD.
68. AII
p3rall, lograms
00
the (ame or equal h(:s,
nnd bellveeo ¡he fame
par~lIc1s,
are equal¡o ooe arolher;
thal is, if BD aod GH
(lig. 5
l.) be equal, aod the lines
1311
and A
1"
be parJlld, th(n Ihe
pmllelogr.m! .~
IIDC,
BDfE. aod EFHG, are eGu. I¡J ooe aoolher. f or AC
is el¡ual to El", <deh being
e~u. 1
lORD, (by
,.r.
l.
of
67.) To bOl h ajd CE, ¡heo AE will
b~
equ.1Cf. So
in ,he ,\VO ¡r;Jngles ABE CD
r,
AIl a
Icg
of
t~e
one,
is equal lO CD a leg in tile other; anJ AE is
c~ ual
10
CF, and Ihe aoglcBAE isequal lOthe aogleDCF (byarto
36.);
,herefore the t',Vo triangles ABE CDF are e–
qll:tl (by arl.'6 \.); aod ¡aking Ihe triangle CKE from
bOth, ¡he (,gure AHKC will be cqual to ¡he figure KDFE
¡
to bo,h IVI,ieh add ¡he li,tle triangleKBD, theo the par,l.
lelogramABDC will becqual to Iheparalldogr3m BDFE.
The (ame way it may be pro\'ed, ¡h.¡ Ihe parallclogram
E.FHG is equal tO ¡he paralldogr<'" EFDB ; fo the
¡hree parallelograms AB DC, BDFE, and EFHG "ill
be equal
ro
one aoolher.
Car.
Henee i¡ is plaio, tha¡ triangl es
ell
¡he (JOle
h.,(, .
and betIVeen Ihe (ame par211c1s, are equal; (,oce ,hey
ar~
¡he h31f uf the pmllelograms 00 the (ame bare aod
be–
tIVeeo the (ame parallds . (by
coro
3.
of lan
arl.)
69.
In an)' ,ighl.ang!d Iriallgi<,
ABC,
(fig. p .)
16, fq/lor,
if
16,
hJp.lhmuft
nc,
.iz.
BCMH,
IJ
T'.!
lo Ih,
flll/J
o¡ Ih, / qua,n IIW!' on Ih, IwaJdN
AB
ani
AC, viz.
lo
ABDE
.nd
ACGF. To dcmonnr"c
thi~,
Ihrough ¡he point A draw AKL perpendicul." ¡O the
hypulhenu(e BC, joio AH, A:,I, OC, and IlG;
Ih~n
i¡ is plain Iba¡ DB is equal tO
BA
(by
art .
5] .). alfo
!lH
is equal to BC (by Ihe fame) ; fo io lhe tilO tri loglcs
DBC
.~B H,
the t\Vo leg! DIl ann BC in ,I:e ol;e '"
cqu;¡J
ro
¡he ,IVO le¡;s AH ,nd
PoI-!
in Ihe o,ha ; ar.o tlle
included ,ngles DBC ann AIlH m al(,. equal; (ror DR .\
is equal
10
CBH.
being bo,h ,igh t; lO eaeh dJJ
ABt'.
,h<n it is pl, in that DIlC is equal lO ABII ) Ihelerore Il.e
triaogks D!l CABlüreequal (byal! .
Ó\. )
bll" lte
I r;ao~'
.
DRC is half of Ihe rqu.. e
tlBll E
(It)'
ror.
, .
of
Vid,)
~nd Ih~
Iri.ln):le AIlH 's hl lf ¡he 1'" . lkl":,. ,,, B¡'; U l
(b)'
Ihe (.,,,,e), ¡hcrd orc half Ihe f'l "a" ,.\1>01:: i¡ q u. 1
10
h¿lf ,he p.,a!ll'lo¡:r,tnl
t: KL.
C:':\(,q'I<'ntly
lI,o!
I~I".II '
ABD F. is equ.d lO Ihe
pJ,.lldo.~r.l nl
B:' f.11.
'1'1:« , 111:
"ay ir may be
pro.ed.lil.II
Ihc
Cq u .r~
:Iec r
'~nl 'J
I
lO Ihe pa¡¿ llclur.r.IIlI KC¡\IL. S" Ih, r"I1I.)llhe
1~11t:1I ,<
/\BDE
ami ACC F
i$
<gil.
.!
,I!:
C""1
,)1'
Ihe
¡". lltI.,
grllllS¡;1\: 1.I·1"rtl 1\(' \11., b'I!
11..
lillll
,)1'
,h&
r.tr..
~
Id~.)~r.lIl1s
is l(l"J I tu t!:c Iquare l:l'\ III,
'::,'''1," :
Ih.·
t
7L
r..
t: