278 Strong’s Problems. 
respectively a cet poe from the points of intersection of 
the given circle and line 
Prosiem XII. 
It is required to describe a circle to touch a straight line 
given in position, and two circles given in position and 
magnitude. 
Case I. When the touching circle circumseribes both 
the given circle and touches ‘the straight line, or ciream- 
scribes neither of the given circles and touches the given 
straight line. 
Const. Let (Fig. 10, pl. 3.) AB be the given straight 
line, GQS, MPO, the given cincles, it is required to de- 
sctibe a circle to touch at B and likewise touch the circles 
et [ and N be the centre of the given cir- 
cles. . ” From I, with radius=radius of QG@S—radius of M 
PO (if GQS> oo ‘eges the circle HRK. Draw 
also the line CD parallel to AB, and distant from it by a 
line=radius of the circle MPO. Then through N describe 
the circle NHF touching HRK in Hand CDin F. Let L 
be thé centre of this circle. From L with radius=radius 
of the circle HFN +radius of the circle mie describe the 
circle EGO, which will be the circle required 
— Demo none Bors oin LHG which will pass through 
fe _ Now because +radius of the cirele MPO and 
{G=HI4-cadis otek the circle MPO, G is in the circumnfer- 
ences of EGO, SGQ. And if at G a perpendicular be 
atin it will touch both circles EGO, SGQ at the point 
G. Therefore these circles touch each other at the point 
G. In like manner it may be shown that the circles EGO, 
MPO touch at the point O. But EGO likewise touches 
the straight line AB. For join LFE. Let this line cut the 
circle EGO in E. Now because LE=LF + radius of the 
circle MPO, the point E falls in the line AB. And be- 
eause AB and CD are parallel ALEA=/LFC. But 4L 
FC is a right angle—(F being the point of contact of the 
eircle FHN and line ED) Therefore LEA is a right an- 
gle, and consequently AEB touches EGO in E, wherefore 
EGO is the circle required. 
By using LE +radius of MPO for LE—radius of MPO, 
this demonstration is applicable to Fig. 11, when neither 
the circles is comprehended. 
