GEOMETRY, 
431 
be fqtiares on AB, AC} then will FG, and GD, 
each equal to C B ; hence F D is a fquare on a line 
equal to C B ; alfo F B and FE a?e redlangles on lines 
equal to C, CB; but tlie fquares AF, FD, and the 
redlangles FB, FE, fill up the fquare AD, or are 
equal to A D. 
118. Corol. I. Hence the fquare of AC, the difference 
between two lines AB, CB, is equal to the fquare of 
the greater A B, leffened by the fquare of the lefs CB, 
and by two reftangles under the fmaller line CB, and 
the faid dilference: 
That is, AB—BC®= AB*—Fc“—aBCXAC. 
For AB^B C=AC. 
Then AC*=AB®—BC*—2BCxAC. 
119. Carol, z. The difference between the fquares on 
two lines A B, AC, is equal to the reflangle under the 
fum A B-J-B C and difference AB—BC of thofe lines. 
That is, AB* —‘aC*=AB+BCXAB—AC. 
For AB*~AC*=.(rB*-f 2^X^. 
=CB 4 -AC+ACXCB. 
120. 
= AB+ACX (CB=:)AB—AC. 
In tvery triangle ADC, the fquare of a fide CD 
j) fubtending an acute angle A, is equal 
to the fquares of the fides A D, AC, 
about that angle., lejfened by two reElan- 
gles under one of thoje fdes AC, and 
that part A B contained between the 
acute angle and the perpendicular D B 
drawn to that fde AC from its op~ 
pofte angle D. 
A B C 
That is,^’* = AD*-fAC* —2CAB. 
DC*—BC*=:(DB*=:) AD*—AB* (xii>. ’ 
And AC*=2ABC4.BC* + AB* (117). 
Then Ic*— AC*=AD*—BC*—2AB*—zABC. 
And DC 
-2 -2 
BC -i-AG . 
:AD* 4 -AC*—2AB*—2ABC, by 
adding 
Then 
——2 -2 —2 
DC =AD +AC - 
(—AB X 2AB—BC X 2 AB) 
(—AC+BC X2AB ) 
-(ACXiAB=)2CAB. 
121. Corol. Hence AR— +AC —DC 
2CA 
J22, In an obtufe-angled triangle ACD, the fquare of the 
fde AD oppofite to the obtife angle 
C is equal to the fum of the fquares 
of the fdes AC, CD, about the 
obtife angle ; together with two red- 
angles under one fde AC, and the 
continuation of that fde to meet 
a perpendicular DB drawn to it 
from the oppofte angle D. 
That is, AD*=:J^*4.CD*+2 ACxCB. 
For AD*—aF*=:(DB*=;)^*— 
And AB*=2ACB-f^*-l-^*. 
Then AD* =AC*4-CD*+2ACB. 
i 123. Corol. Hence , 
2AC 
124. In any circlef a diameter, AB, drawn perpendicular 
to. a chord DE, bifeBs that chord and 
its fubtended arc DBE.^—From the 
centre C, draw the radii CD, CE, 
to the extremities of the chord DE ; 
then the triangles CFE, CFD,' are 
congruous (102). For CF being 
at right angles to DE, the CFD 
= ^CFE; and the triangle CDE 
being ifofceles (23), the ^D = ^E 
(104). Alfo CF is common; there- 
fore DE^FE, and the arc DBizzBE ; B 
for thofe arcs ineafure the equal angles FCE, FCD. 
125. Corol. Hence, in a circle, a right line draW'n 
through the middle of a chord at right angles to it, 
palTes through the centre of that circle} and the con¬ 
trary. 
126. A tangent AB, to a circle, it perpendicular to a dia<- 
meter DC, drawn to the point of cort- 
taB C.—If it be denied that DC ^ 
is perpendicular to AB, then from 
the centreD,let fome otherlineDB, 
cutting tlie circle in F, be drawn 
perpendicular to AB. Now the 
angle DBC being right, the angle 
DCB is acute; confequently DC 
is greater than DB. But I)C=: 
DE (9); therefore DE is greater 
than DB, whiclt is abfurd. Therefore no otlier line 
palling through the centre can be perpendicular to the 
tangent, but that which meets it at the point of contact. 
127. An angle BCD, at the centre of a circle, is double of 
the angle BAD, at the circumference, 
when thofe angles fand on the fame 
arc BD.—Through the point A 
draw the diameter AE; then the 
angle ECD —^ CAD-f-^ CDAj 
but the ^CAD=;^CDA ; there¬ 
fore the A. ECD is equal to twice -E 
the angle CAD. In the fame 
manner it may be fhewn, that 
the angle BCE is equal to twice 
theangleBAE; confequently tlie angle BCD (=^ECD 
-|-^BCE) is equal to twice the angle BAD (z=:^EAD 
+ZBAE). 
C/i 
128. Carol. I. Hence an angle, B AD, at the-circum¬ 
ference, is meafiired by half the arc, BD, on wliich it 
Hands; for the angle at the centre BCD is meafured by 
the arc BD ; confequently the angle BAD — half the 
angle BCD, is meafured by half the arc DB. 
129. Corol. 2. All angles in the circumference, and 
Handing on the fame arc, are equal. 
130. An angle, BAC, in afmicirde, is a right one-. An 
angle, DAC, in a fegment lefs than 
a femicircU, is obtife. An angle, 
E AC, iu a figment greater than a 
femicircle, is acute. —For the angle 
BAC is meafured by half tJie 
femicircle arc BEC, or is meu- 
fured by half of 180°; that is, 
by 90”. And ^DAC is mea¬ 
fured by half the arc DEC, 
greater than 180°; alfo _^EAC is meafured by half the 
arc EC, lefs than 180°; therefore thefe angles are re- 
fpecbtvely equal to, greater, or lefs, tliaii 90^. 
131. Cor. Hence in a right-angled triangle BAC,, 
the angular point A of the right angle, and the ends 
B, C, of the oppofite fide, are equally dillant from F, 
the middle of that fide; that is, a circle will always 
pafs through the right angle, and t-he ends of its opp'o- 
Hte fide taken as a diameter. 
4 
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Th^ 
