, 
56 Strong’s Problems. 
Through the points C, D, E, describe the circle CDE, 
which hall be the vircle required. 
Demons. For since BE = mean proportional between . 
- BD and BC, BE2e= BD. BC. Now since the circle passes 
through the points, D, E,and BD. BC = BE’, o straight 
line BA touches the circle. (Euc. TIL. 37.) Q. E 
Cor. to Case I. Ifthe pomt D should fall in EC pro- 
duced, bisect the distance between the two points, and the 
proof is as before. 
Prosiem IV. 
Let two straight lines and a point which does not lie at 
the intersection of those lines, be given in position, it is re- 
quired to describe a circle through the given point to touch 
the two given straight lines 
Case 1. When the given point lies in one of the given 
straight lines. 
Construction. Let AB, AC 
be the given straight lines, andD 
the hee point in one of the lines. 
Let the lines produced if necessa- 
ry meet at A. Bisect the angle 
BAC by the straight line AE. 
Through D draw DE at right an- 
gles to AC, cutting the bisecting 
line-in E. From E as centre w ith 
ED as radius describe a circle : 
which shall be the circle required. 
—For draw EF at right angles to AB. 
Demons. ‘The angle FAE = angle DAE and angle 
AFE = angle ADE and the side AE is can to both 
the triangles AFE, ADE. Therefore EF = ED. There- 
fore a circle described from E as centre with Se as radius, 
will pass through F. Now EF and ED are at right angles | 
to AB and AC. Therefore the circle touches AB and AC 
in Fand D. And (by Const.) it passes through D.—Q. 
f > 
Case HI. When the point is upon neither of the lines. 
Const. Let AB, AC (fig. 6.) be the given straight lines, 
on = the given point. Let AB, AC , produced if necessary, 
meet in A. Bisect the angle BAC by the straight line AE. 
pare in 
