Let O be the center of the circle with radius R and Oi the center of
circle Ci. Congruent isosceles triangles O1OO2, O2OO3, ... ,
On-1OOn,
and OnOO1 about point O imply
∠O1OO2 = 2*pi/n and ∠OO1O2 = pi/2 - pi/n.
Applying the law of sines to ΔO1OO2,
|O1O2| |OO2|
--------------------- = ------------------
sin(∠O1OO2) sin(∠OO1O2)
2r r+R
--------------------- = ------------------
sin(2*pi/n) sin(pi/2 - pi/n)
2r r+R
--------------------- = ------------------
2sin(pi/n)cos(pi/n) cos(pi/n)
r sin(pi/n) 1
--- = ------------------ = ---------------
R 1 - sin(pi/n) csc(pi/n) - 1
QED
Note: n=6 ⇒ r/R = 1. Which is what one would
expect after arranging six pennies about a
seventh.
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