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If two circles of radius a and b unit touch externally and a circle of radius c unit touch these two circles n also to common tangent to to these circles .prove that one upon root of c equal to sum of one upon root of a and one of root of b

If two circles of radius a and b unit touch externally and a circle of radius c unit touch these two circles n also to common tangent to to these circles .prove that one upon root of c equal to sum of one upon root of a and one of root of b

Grade:10

1 Answers

Vikas TU
14149 Points
4 years ago
Assume circles as, the red circle touches the other two circles as well as their common tangent BD.
Let AF=BD=d,
BC=x and CD=d-x,
d2 = (a+b)2 – (b-a)2=4ab (apply pythogorous theorem in triangle AEF)
d=2√ab1
similarly in triangle AHG,
x=2√ac2
Similarly in triangle GEI,
d-x = 2√bc3
from 1,2,3 
2√ab=2√ac+2√bc4
Dividing 4 by √abc, we get
1/√c=1/√a+1/√b
 

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