_Strong’s Problems. BIG 
Case 1]. When the touching circle comprehends one of 
the given circles. 
Const. Let (Fig. 12. pl. 3.) AB be the given straight 
line and QHP, LRS the given circles, of which LRS is to 
be comprehended by the touching circles. Let G, K be the 
eentres of the given circles respectively. From G with ra- 
dius equal to the radius of HPQ+radius of LIS describe 
the circle NM{. Draw also CD parallel to AB and distant 
from it by a line==radius of the circle LRS. Through K 
describe the circle KFI touching NIM inland CD in F, 
of which circles let O be the centre. Increasing the radius 
by a line=radius of the circle LRS describe the circle LE 
H, which will be the circle required. 
Demonstration. For join OIH, OKL, OFE : let OLH 
cut QPH in the point H. Now because GH =OI + radius 
of the circle LRS, H is the circle LHE; and if at the point 
H aline be drawn perpendicular to OH it touches the circle 
LHE at the point H, it will likewise touch QHP in H. 
For if the line OH be extended it will pass through G the 
centre of QHP. Therefore QHP, LEH itouch each other 
at H. In like manner LRS, EH touch each other at L. 
But the circle LEH touches AB. For let CFE cut EHL. 
in E. Now the angle OFC being right OF will cut AB at 
right angles. And because OK=OF-+ radius of LRS, E 
falls in the straight line AB. But it has been proved that 
OFE cuts AB at right angles. Therefore AB touches EH 
LatE. Wherefore EHL is the circle required. 
_ Note.—The circles must always be on the same side of 
the line ; when they cut the line the touching circle cannot 
eircumscribe them. When one of them lies wholly within 
the other the given line must cut one or both of them: oth- 
erwise the problem is impossible. 
Prosxiem XIII. 
There are two points and a straight line given in position, 
it is required to draw from the points to a point in the 
straight line, two lines whose differences shall be equal to a 
given line. 
Const. Let (Fig. 13. and 14. pl. 3.) AB be the given 
line, C, D the given points and X the given difference. 
From C as centre with radius=X describe the circle LIO. 
