Strong’s Problems. 271 
‘Const. Let (Mig. 9. pl. 2.) ALA’, BMB’, CNC’, be 
ihe given. circles, of which ALA’, BM’, are equal, and 
CNC’ is less than the othertwo. Let G, H be the centres 
of ALA’, BMB’, From G, H, as centres, with radius =ra- 
dius of ALA~radius CNC, describe the circles DD’, EE’. 
Through F the centre of the circle CNC’, describe the ecir- 
cle FDE, (Prob. TX.) touchng DD’, EE’ in D, E, of 
which circle, let O be the centre. Join CD, and it will pass 
through G. ‘Then from O as centre, with radius =OD+ 
CF (=radius of the circle CNC’) describe the circle ABC, 
which shall be the circle required. 
Demonstration. Fer join ODA. Now becaiee ODA= 
radius of the circle ABC’ (=OD+DA, or FC) of which 
O is the centre, A is in the circumference of ABC. And 
because ODA passes through G, and GD+DA=radius of 
the circle ALA’, A is in the circumference of ALA’. Hence 
ALA’, ABC meet in A. And they likewise touch in A. 
For if Av be drawn at right angles to ADO, A will be a 
tangent to both circles in the same point A. Whence the 
circles must likewise touch in that point. In like manner 
it may be proved that ABC, BMB’, ABC, CNC’, touch 
each other respectively at the points B,C. ABC is there- 
fore the circle required. 
Now by using OD’/— AD’ for OD+ AD, and A’, BY, &c. 
for A, B, &c. the demonstration is the same when none of 
ihe given circles is comprehended. 
2. When the touching circle comprehends both the equal 
circles, and touches the ‘smaller one externally 3 or compre- 
hends the smaller circle, and eae the equal circles ex- 
ternally. 
Construction. Let (fg. 10. pl. 2.) Gy, Fa, Tz, be the 
given circles of which Gy=Fr. Let A, B, C be the cen- 
tres of these circles respectively. From A, B, as centres 
with radius—radius of the circle Gy-+- radius of the circle 
{z describe the circles De, Fn. And through C, the cen- 
tre of the smaller circle, describe a circle CDE touching 
the circles Dg, F'n, inthe pomts D, E, of which cirele let 
O bethe centre. From O with radius—radius of the cir- 
cle CDE —radius of the circle Iz, (i. e. OD—GD, or CI,) 
' describe the circle GIF, which shall be the circle required. 
Demonstration. Yor join OD, OE, which will pass 
through ¢ centres A,B. Now (by Const.) D is in the 
