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Strong’s Problems. 55 
Demonstration. Because the angle FCD = angle GCD 
and the angle DF = angle DGC, and DC is common to 
both the triangles DFC, DGC, the straight line DF 
straight line DG. In like manner it may be shewn that 
== DE. Therefore a circle described from D as cen- 
tre with DF as radius will pass through the three points E, 
, G. And it is manifest also that it ‘touches the lines H, 
AC, CB, in those points, since the radii DE, DG, DF are 
severally Seeaantiocier to the lines BH, BC, CA. -Q. Eu. 
Prosiem IIL. 
Given two points and a straight line in position, the 
‘points not being on opposite sides of the line ; it is require 
to describe a circle the circumference of which shall pass 
ee the two given = ote and touch the given line. 
Case I. When one of the given points is in the given 
steuiene ht lin 
~ Const. Ea AB i e 3.) be the given straight line, C 
the given point in AB, and D the other given point.—Join 
DC, and through C draw CE at right angles to AB. At 
the point D in the line DC, make the angle CDE = the an- 
gle DCE. Then the side DE = side CE. Therefore a cir- 
cle described from E as a centre with radius DE, will pass 
through C, and D. And it will likewise touch the line 
AB, this line being perpendicular to the radiu 
Case H. When the straight line joining tis two given 
points is parallel to the given straight line. 
Const. Let AB, (Fig. 4.) be the given straight line, and 
> D, the two given points. Join d bisect CD in 
F. From F draw F E, at right anglestoCD. Let FE ex- 
tended cut AB in E. Through C, D, E describe a circle, 
Which shall be the circle taste 
emonst. For Join ED, and EC. Because the angles 
EFC, EFD are equal, and chs FD and FE is common, 
the angle’ FCE = angle FDE. But the angle FCE= 
alternate angle CEA. Therefore CEA = CDE. Therefore 
AB touches the circle CDE in the point E. (Eucl. If. 22.) 
Case If]. When the straight line joining the two given 
points is oblique to the given line. 
Const. Join CD (Fig. 5.) and let CD produced meet AB in 
B. Take SBE Ss == a mean proportional between BD and BC. 
