Strong’s Problems. 267 
segment RBH, and the angle IHO=RHM=anele in the 
segment HLKI. But the angle in the segment RBH= 
eae in the een IDEP. ‘Therefore the angle in the 
or NIP. ae the angle in the segment (DEP—NIP; 
wherefore NQ touches DEPI in the point I. Consé- 
quently, HLKI touches IDEP in I. Now (by Const.) 
HILKI touches RBH, and passes through L. Wherefore 
it is the cirele required. | 
Case If. When the two circles are unequal, and the cir- 
éle which passes through the given you circumscribes 
them. 
Const. Let (Fig. 2. pl. 2.) as in Case i. the point be 
Li, and the circles HBC, DGE. Draw the tangent FG, 
and extend it till it meet cy, produced in A. Let xy pro- 
duced cut the given circles in C, B, E, D. Through L, 
and C, D, the remote points in which avy cuts the given cir- 
cles, describe (Prob. i.) the circle LOD. Suppose AL 
produced to meet the circumference of this circle in K. 
Through the points, K, L, describe (Prob. v.) a circle 
touching HBC, in H. Then shall this be the circle re- 
quired. 
The points AI’, AT being joined as in case I. and the tan- 
gents MO, NQ being diawn, the demonstration employed 
in case It, is applicable to this. 
Case If. When the touching circle circumscribes one 
of the given circles, and touches the other externally. 
Const. Let (Fig. 3. pl. 2.) HBC, and HGE be the 
given circles, and Li the given point. Join xy, the centres 
of the circles, and extend this line till it cut the circumfer- 
ences of the circles in B,C, D, E. Draw FG, a tangent to 
the circles in the points F, G. Let this cut the line, CE 
in A. Join AL. Through L, ©, D, describe the circle L 
CD. Suppose AL extended cuts this circle in K. Through 
L, K, describe the circle LKH touching BHC in H. Join 
AH, and let A H produced cut DGE in L, and HKL in 
I’, Draw, as in cases I and IT, MHO touching the circles 
AKL, BHC, in H, and NQ touching HKI’L in V. 
Now by applying the Demonstration in case I, the circle 
HKI'L, as in former cases will be found to answer the con- 
ditions of the problem. 
a\ 
