: Strong’s Problems. 20% 
cles HGL, DIE meetin K. They also touch in that point. 
For if they do not they must meet in some other point; let’ 
them meet ine. Join Dx. Let Dx produced cut HGL in” 
yand ABiny. Then Dy, De=LD, DK. But DzE be-. - 
ing aright angle as before, Dry, Dz =FD, DE=LD, DK. — 
Therefore Dx, Dy =D, Dz and Dy=Dz which is’ Hien 
Therefore the circles do not meet inv. Now « being any 
point they meet in no point but K. They therefore fouch 
in K. Wherfore LGKH is the circle required. 
Case IIL. When the given circle cuts the given straight 
line. 
Const. Let (fg. 9. pl. 3.) AB, be the given straight 
line, H the given point, and FD’E the given circle. Let 
the circle FD’/E cut AB in L, M. Through C the centre 
of the given circle draw Cl at right anglestoABin I. Let 
this line produced cut the circle FD’'E in E, F. Through 
H, 1, E deseribe the circle HIF. Join EH. Let this pro- 
duced cut G, HIF inG. Through H, G describe the circle 
GHOD touching AB in O. And this will be the circle re- 
quired. 
Demonstration. For jom OE, let EO extended cut E 
DF in D’ and GHOD in D. Then (E. 6. p. I.) EO, E 
D’=El, FE=EH, EG=EO, ED. Therefore since EO, 
ED= EO, ED’, ED= ED’, and the circles FD’ E, GHOD 
meet in D. But they also touch in this point. For if not, 
Jet them as before meet in z. Join Ex cutting AB in y, 
and GHOD in z. Then joining Fa, the angle EF is a 
right angle. Therefore EI: Ez: : Ex: EF wherefore E - 
1, BE =fz, Ex. But == EF= EO, ED=EH, EG=Ey, 
Ee. Therefore Ex, Ez=Ex, Ey, and Ez —Ey which is 
absurd. Therefore tie Sand FDE, GHOD do not meet 
inw. Now a being any point but the point D, they meet in 
no point but D. They therefore touch in D. Therefore 
GHOD is the circle required. | 
Note-—Whien the circle does net cut the line, the point 
must be given without the circle and on the same side of 
the line with the circle. When the circle cuts the line the 
point may be given any where except at the point of inter- 
section of the line and circle. If the circle to be described 
‘is to touch the given circle externally, the point may be giv- 
en any where without the circle or in the circumference, — 
except in the points EH, F, (see Fig. Case Ill.) which are 
Vor. Wl No. 2, 36 
