58 _ Strong’s Problems, 
that it touches the given circle. For draw LX touching the 
given circle in the pomt B. Then FB will be at right angles 
to LX. Now LX being at right angles to BE at the pot 
B, which is in the circumference of the circle CBM, must 
touch this circle at that point. ‘Therefore since both circles, 
AB, CBM, touch LX at the same point B, they must touch 
each other at that point. 
In like manner by using the letters C:, D-, E:, &c. for 
C, D, E, &c., the demonstration will apply to the case, 
where the point is within the circle. 
Case II. When the two points are either without or with- 
in the given circle at unequal distances from the centre. 
Const. Let ABD (Fig. 8.) be the given circle, and C, E, 
the two poimts without the circle. It is required to describe a 
circle through those points which shall touch the given cir- 
cle—Take any point X within (or X- without) the given 
circle which is not in the same straight line with CE. And 
through the points C, E, X, describe a circle.-—(Prob. I.) 
- Let this circle cut the given circle in the points B, D. Join 
BD; and through the points C, E, draw CE meeting BD ex- 
tended in F. Through F, draw FA, touching ABD in A. 
(E. ILI. 17.) : 
Demonst. Because the straight line FD cuts the circle 
ABD, and the straight line F'A touches it, FD. FB=FA?. 
But FD. FB = FC. FE. Therefore FC. FE = FA?. Let 
therefore, a circle be described through (Prob. I.) C, E, A. 
—Now this circle meeting FA in A, and FC, FE equaling 
FA?; FA must bea tangent to CAE, at the point A. Since, 
ilrerefore, both the circles, ABD, CAE, touch the straight 
line F'A at the point A, they must touch each other at that 
ot. 
» In like manner, by using, C:, E:, &c. for C, E, &c. this 
demonstration is applicable to the case where the Lypigl are 
within the given circle. 
Case [11. When the two points are either within or 
without the given circle at equal distances from the centre. 
Construction. Let AB (Fig. 9.) be the given circle, and 
C, D, the given points without (or C-, D-, within) the given 
cirele at equal distances from the eeatve: Join CD, ORL 
HD. Bisect CD in E,and jom EH. Let EH cut the cir- 
cumference of the given circle in A. Through the points A, 
