270 Strong’s Problens. 
circumference of the same circle. But the circle DEE 
likewise touches the three given circles. For at the point 
F, draw G, H at right angles to OF, and it will be a tangent 
to the circles DEF, F, F’ at the point F. Therefore the 
circles touching the line GH at the same point F, touch 
each other at that point. In like manner it may be proved, 
that the circle DEF touches the other two circles at KE, D, 
respectively. 
Now by using OA—AD for OA+AD, &c. and D’, EK’, 
FE’, &c. for D, E, F, &c. this construction and demonstra- 
tion are applicable to the case in which none of the given 
circles are comprehended by the touching circle. 
Case If. When the circles are equal, and the touching 
cirele circumscribes one, and touches two, or circumscribes 
two and touches the other. 
Const. Let (Mg. 7. pl. 2.) Ey, Dx, Fz be the given 
circles of which A, B, O are the centres. From O, the 
centre of the circle Ez, describe the circle GLM, witls ra- 
dius=Q radius of the given circles. Through the points 
A, B, describe the circle ABG touching GLM (Prop. v.) 
in the point G. Let C be the centre of ABG. Then from 
C as centre, and CG—FO as radius, describe the circle D 
EF, which shall be the circle required. 
Demonstration. For join C, O the centres of the circles 
ABG, GLM. Extend co, and it will pass through the point 
of contact of these circles. Joinalso CEA, CDB. Now 
because CEH =CA—AE the radius of the given circles, E 
is in the circumferences of the circles Ey, DEF, and if at 
the point & a line be drawn at right angles to CA, it will be 
a tangent to the circles Ey, DIcF at the same point E. 
Therefore these circles touch each other at the point E. 
Tn like manner it may be proved that the circles Dr, DEF 
touch each other in the point D, and that Fz, DEF touch 
each other in the pomt EF’. ‘Therefore DEF is the circle 
required. 
By using CA4- AE, &c. for CA—EA, &c. This demon- 
stration is applicable to the Fig. in which the touching circle 
comprehends two of the given circles, and touches the 
other. 
Case If. When two of the circles are equal. 
1. When all the circles or none of them are compre- 
rended. 
