58 Strong’s Problems. 
that it touches the given circle. For draw LX touching the 
given circle in the point B. Then FB will be at right axles 
to LX. Now LX being at right angles to BE at the point 
8, 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, Se. for 
C, D, E; &c., the demonstration will apply to the case, 
where the point is within the circle. 
ase II. When the two points are either without-or with- 
in the givén circle at oie distances from the centre. 
the two points 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 srt —(Prob. I.) 
_ Let this circle cut the: given Nacho 3 in the points B, D. Join 
BD; and through the points C, E, draw CE mae BD ex- 
tended in F. Through F, draw FA, — ABD i in A. 
¢ ERE: T%: 
Demonst. Because the straight lng FD cuts the mains 
ABD, and the straight line FA touches it, FD. FB=FA 
But FD. FB=FC. FE. Therefore FC. FE= FA?. Lei 
therefore, a circle be described through (Prob. IL.) C, E, A. 
—Now this circle meeting FA in A, and FC. FE equaling 
FA?; FA must be a tangent to CAE, at the point A. Since, 
ther efore, both the circles, ABD. , CAE, touch the straight 
line FA at the point A, they must touch each other at that 
oint. 
In like manner, by using, C*, Ev, &c. for C, E, &c. this 
demonstration is applicable to the case where the points aré 
within the given circle 
Case Ill. When the two points are either within or 
without the given circle at equal distances from the centre. 
onstruction. Let AB (Mig. 9.) be the given circle, and 
C, D, the given points without (or C-, D-, within) the given 
circle at equal distances from the centre. Join CD, CH, 
HD. Bisect CD in E, and join EH. — Let EH cut the cir- 
cumference of the given circle in A. Through the points A, 
