270 Strong’s Problems. 
circumference of the same circle. But the circle DEF 
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 FY, touch 
each other at that point. In like manner it may be proved, 
that the circle DEF touches the other two circles at E, D, 
respectively. » 
Now by using OA—AD for OA+AD, &e. and D’, E, 
F’, &c. for D, , &c. this construction and demonstra- 
tion are applicable to the case in which none of the given 
circles are comprehended by the touching cirele. 
Case hen the circles are equal, and the touching 
circle circumscribes one, and touches two, or circumscribes 
two and touches the other. 
Const. Let (Fig. 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 Fz, describe the circle GLM, with ra- 
dius=Q radius of the given circles. Through the points 
A, B, describe the circle ABG touching GLM (Prop. v-) 
in the pointG. Let C be the centre of ABG. Then from 
as centre, and CG—F0O 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. Join also CEA, CDB. Now 
because CE=CA—AE the radius of the given circles, E 
is in the circumferences of the circles Ey, DEF, and if at 
the point E a line be drawn at right angles to CA, it will be 
a tangent to the circles Ey, DEF at the same point bE. 
Therefore these circles touch each other at the point E. 
n like manner it may be proved that the circles Dx, DE 
-touch each other in the point D, and that Fz, DEF touch 
each other in the point F, Therefore DEF is the circle 
“required. 
By using CA+AK, &c. for CA—EA, &c. This demon- 
stration 1s applicable to the Fig. in which the touching circle 
seen two of the given circles, and touches the 
® ‘ 
er. 
Case IIT. When two of the circles are equal. 
1. When all the circles or none of them are eompre- 
hended. 
