56 Strong’s IPPoblents. 
Through the points C, D, E, describe the circle CDE, 
which shall be the circle required. 
Demons. For since BE: = mean proportional between 
BD and BC, BE:-= BD. BC. Now since the circle passes 
through the points C, D, E,and BD. BC = BE”, the straight 
line BA touches the circle. (Euc. IT. 37.) Q. ELL. 
Cor. to Case I. Ifthe point D should fall in EC pro- 
duced, bisect the distance between the two points, and the 
proof is as before. 
Prosuem [V. 
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. wi 
Construction. Let AB, AC 
be the given straight lines, and D 
the given 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.3 
Through D draw DE at right an- 
gles to AC, cutting the bisecting 
line in E. From E as centre with 
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 common to both 
the triangles AFE, ADE. Therefore EF —ED. There- 
fore a circle described from E as centre with ED 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 F and D. And (by Const.) it passes through D.—Q. 
Deas & : 
Case I. When the point is upon neither of the lines. 
Const. Let AB, AC (Fig. 6.) be the given straight Imes, 
and D the given point. Let AB, AC, produced if necessary, 
meet in A. Bisect the angle BAC by the straight line AK. 
