A circle centered at A with a radius of 1 and a circle centered at B with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
[asy]
size(220);
real r1 = 1;
real r2 = 3;
real r = (r1*r2)/((sqrt(r1)+sqrt(r2))**2);
pair A=(0,r1), B=(2*sqrt(r1*r2),r2);
dot(A); dot(B);
draw( circle(A,r1) );
draw( circle(B,r2) );
draw( (-1.5,0)--(7.5,0) );
draw( A -- (A+dir(210)*r1) );
label("1", A -- (A+dir(210)*r1), N );
draw( B -- (B+r2*dir(330)) );
label("4", B -- (B+r2*dir(330)), N );
label("A",A,dir(330));
label("B",B, dir(140));draw( circle( (2*sqrt(r1*r),r), r ));
[/asy]
<spanclass=′latex−bold′>(A)</span> 31<spanclass=′latex−bold′>(B)</span> 52<spanclass=′latex−bold′>(C)</span> 125<spanclass=′latex−bold′>(D)</span> 94<spanclass=′latex−bold′>(E)</span> 21 Pythagorean Theoremgeometry