Strong’s Problems. 59 
C, D, describe the circle ACD, which shall be the circle 
required. Through A draw FG perpendicular to EH. 
Demons. Because CE = ED, HC = HD and HE is 
common to the triangles HED, HEC, the angles CEH, 
DEH are equal, being opposite equal sides. Therefore 
HE is perpendicular to DC. Now because CD is a chord in 
the circle CAD and is bisected at right angles by AE, AE 
passes through the centre of the circle. But FG is at right 
angles to EA, and EA passes through the centre of the cir- 
cle CDA ; therefore FG touches the circle CDA in the 
point A. "But (by Const.) FG touches the circle AB in the 
point A. ‘Therefore the circles CDA, AB, touch each other 
at the point A.—Q. E. I. 
In like manner by using the letters C-, D-, &c. for C, D, 
&c. the above demonstration is applicable to the case where 
the points are within the circle at equal distances from the 
centre. 
 Scholium. As CD, GF are both at right angles to EH 
they are parallel to each other. Therefore the construction 
in Case II, failing, Case III is necessary. 
Note I. When one of the points is within the circle and 
the other without, the problem becomes impossible ; for 
then the circle which passes through those points will cut 
the given circle, which is against the Hypothesis. 
Note II. All the cases of this problem (except the first) 
admit of two polanons> 3 as is manifest from the above con- 
struction. 
Prosiem VI.. 
It is required to describe a circle to touch two given 
straight lines and a given circle. 
Case I. When the two given straight lines are parallel 
and the given circle lies between them, or cuts one or both 
of them. 
Const. Let AB, CD (Fig. 10.) be the two given straight 
lines, and MI the given circle. Draw EF’ parallel to AB 
and distant from it, by a line = radius of the given circle. 
Draw also GH parallel to CD and ata like distance from CD. 
It is here to be noted that if EF fall between the given lines 
GH must likewise. Through Q the centre of the given 
circle describe the circle QNS touching EF, GH in N, 8, 
