help with geometry

Here's one way to  solve this....but....maybe not the fastest....!!!!

Let D  =(0,0)    E  =  (0, 1/2)    B  =(1.1)  C  = (1,0)

Call  the center of the circle  ( m, 1/2)

So  the distances from each of these points to the center  are equal  = the radius

We have this 

Distance from center to E   = distance from center to B

( m - 0)^2  + ( 1/2 - 1/2)^2  =  ( m - 1)^2 + ( 1/2 -1)^2

m^2  =  m^2 - 2m + 1  + 1/4

2m = 5/4

m = 5/8

So  the radius  =   the distance from (m,n)  to E  =  

sqrt  [ 0 - 5/8)^2  +  ( 1/2 - 1/2)^2 ] =

sqrt[  ( 5/8 )^2 ] =  

5/8

ABCD is a square with a side length of 1.  A circle passes through B, C, and is tangent to the midpoint of AD (at E).  What is the radius of the circle?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

AB = EF = 1      BF = 1/2         ∠BEF = tan^-1(1/2) = 26.56505118º

EB = sqrt(EF² + BF²) = √1.25

Radius      EO = 1/2 (EB / cos ∠BEF) = 0.625     or   5/8