Circles C₁, C₂, ... C_n ( n≥3 and each with radius r ) are externally tangent
to a circle of radius R. Find the ratio r/R (in terms of n) if each of the
n circles is externally tangent to both of its neighbors.

Another way to look at it is that the outer circles are at the vertices of a cyclic polygon with n sides.

Wikipedia gives the circum radius of such a polygon as P=s/(2*sin(pi/n) 

The radius of each outer circle is clearly  r=s/2.   
Hence the radius of inner circle R=s/(2*sin(pi/n)-s/2
and r/R=(s/2)/(s/(2*sin(pi/n)-s/2) 
This simplifies, irrespective of the length s of a side, to the solution:  

r/R = 1/(csc(pi/n)-1) 

Note: checking against Steve's solution, it is indeed true that: sin(pi/n)/(1-sin(pi/n)) = 1/(csc(pi/n)-1)

 

Edited on August 8, 2013, 1:41 am

Posted by broll on 2013-08-08 01:33:44