Strong’s Problems. 275 
Demonstration. For join ODA which will pass through 
the centre B. Now because OA=OD+ EM and BA=D 
B+EM, (if M be the point in which a line joining O, E, 
cuts the circle FM) A is in the circumferences of AL, AFH. 
ndifat A a perpendicular be erected both circles will 
touch it at the same point A. Therefore they touch each 
other at that pee In like manner it may be shown that 
NH, AFH, MF, AHF respectively eae at the points H, 
E. Therefore AFH is the circle requir 
By using OD—EM for OD+EM this demonstration is 
» applicable to fig. 4, in which noue of the circles are compre- 
hended by the given circle. 
2. When the touching circle comprehends and touches 
one externally or comprehends one and touches two exter- 
nally. 
‘ens Let (Fig. 5. pl. 3. ) HK, ya: LE, be the xiv- 
en circles whose centres are A, B, C. Let tE be the cir- 
cle which is not to be eircamstribed alone. From A with 
vadius=radius of HK + radius of LE describe the circle D 
1. And from B with radius of MG,+radius of LE, de- 
scribe the circle FN. Through C describe the circle DCF 
touching DI, FN in D, F,—of which circle let O be the 
centre. Decreasing the ace? of this circle by a line=ra- 
dius of the circle LE (or in Fig. 6- increasing it by the same 
line) describe the circle HEC. Shick shall be the circle re- 
quired. 
Demonstration. For join OC and it will cutthe circle LE 
in the point E. Because OE—OC—CE, E is in the cir- 
cumference of HEG. Therefore if from Ea perpendicu- 
lar be erected, it will touch both circles in the same point 
E. Therefore they touch each other in that point. In like 
manner the circles G, M, HEG ; HK, HEG, touch re- 
spectively at H, G. 
Now by using OC +CE for OC—CE this i: ae gee 
applicable to Fig. 6. pl. 3. in which one circle is com 
hended and the ather two touched externally. 
Proscem XI. 
lacs screquired to draw a circle through a given point, te 
touch a straight line given in position and a circle given in 
magnitude and position. 
