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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.

 Dec 8, 2019
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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

 

 

cool cool cool

 Dec 8, 2019
edited by CPhill  Dec 9, 2019

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