O74 -  Strong’s Problems. 
demonstration answers for Figure 16, in which the larger cir- - 
cle is alone comprehended. 
6. When the touching circle comprehends one of the 
equal circles, together with the larger, or one of the equal 
circles alone. i 
Const. Let (fg. 1. pl. 3.) HPN, MGQ, DER be the 
given circles of which DER=MGQ. Let A, B, C, be the 
centres of these circles respectively : From A with radius 
=radius of the circle HPN-+ radius ofthe circle DER de- 
scribe the circle IK: and from B with radius=two radii of 
the circle MGQ describe the circle FL. Through C de- . 
seribe the circle CFI touching IK, FL in the poits I, F ; 
of which circle let O be the centre. Decreasing the radius 
by a line=radius of the circle DER describe the circle HE 
G which shall be the circle required. 
Demonstrations. For join OC. Let OC cut the eircle 
DER in the point E. Now because OE—OC—CE, E is 
m the circumference of the circle HEG. If therefore (Fig. 
2. pl. 3.) as in former cases, a perpendicular be erected at 
E, it will touch both circles at that point. Therefore the 
circles DER, HEG touch each other in the pomt E. In 
like manner it may be shown that the circles HPN, HEG; 
GQM, HEG respectively touch at the points H, G. There- 
fore HEG is the circle required. 
By usng OC+CE for OC—CE this demonstration is 
applicable to Fig. 2, in which the touching circle compre- 
hends one of the smaller circles and touches the other, to- 
gether with the larger circle externally. 
Case IIi. When all the circles are unequal, 
1. When the touching circle comprehends all or none of 
the given circles. 
Cons. Let (Fug. 3. pl. 3.) AL, HN, MF be the given 
circles of which FM isthe least. Let B,K, E, be three cen- 
tres. From-B describe the circle DG whose radius=radius 
of the circle AL—radius of the circle FM. From K de- 
scribe the circle IP, whose radius=radius of the circle NH 
—radius of the circle FM. Through E describe the circle 
EDI touching DG, PI in D,J. (Prob. V.) Let O be the 
centre of this circle. From O, with radius=radius of the 
circle DEI+ radius of the circle FM, describe the circle 
AFH, which shall be the circle required. 
