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The radius \(r\)  of a circle inscribed within three mutually externally tangent circles of radii \(a\)\(b\), and \(c\) is given by

\(\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.\)
What is the value of  \(r\) when \(a=4 \)\(b=9\) and \(c=36\)?


 

 Dec 29, 2023
 #1
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1/r =  1/4 + 1/9 + 1/36 + 2sqrt [ 1/36 + 1/144 + 1/324 ]

 

1/r =  14/36  + 2sqrt [49/1296]

 

1/r = 14/36 + 2 (7/36)

 

1/r  = 14/36 + 14/36

 

1/r = 7/9

 

r = 9/7

 

cool cool cool

 Dec 29, 2023

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