The length of a radius of a circle is 3 + 2*sqrt(3). Three congruent circles are drawn in the interior of the original circle, each internally tangent to the original circle and externally tangent to the others. Find the length of a radius of one of the three congruent circles.
The centers of the circles will form and equilateral triangle each with a side of 2r
The circumradius of this triangle = (2r) / √3
And the circumcenter of this equilateral triangle will be the center of the larger circle
So.......the radius of the larger circle = r + (2r)/√3 = [ 3 + 2√3] r
_________
3
So we have that
3 + 2√3 = [ 3 + 2√3] r
__________
3
1 = r / 3
r = 3 = radius of smaller circles
Here's a pic :
DE = radius of smaller circles