Three circles with radius `r_(1), r_(2), r_(3)` touch one another externally. The tangents at their point of contact meet at a point whose distance from a point of contact is `2`. The value of `((r_(1)r_(2)r_(3))/(r_(1)+r_(2)+r_(3)))` is equal to

Answer

4

Explanation

Three circles ……….
`a = r_(2)+r_(3), b = r_(3)+r_(1) , c = r_(1)+r_(2)`
We have given `ID = IE = IF = 2`

`2 = ("Area of" Delta ABC)/("semi perimeter of" Delta ABC)`
`Delta = sqrt(s(s-a)(s-b)(s-c)) = sqrt((r_(1)+r_(2)+r_(3))r_(1)r_(2)r_(3))`
`2 = (sqrt(r_(1)r_(2)r_(3)(r_(1)+r_(2)+r_(3))))/((r_(1)+r_(2)+r_(3))) = sqrt((r_(1)r_(2)r_(3))/(r_(1)+r_(2)r_(3)))`
`implies (r_(1)r_(2)r_(3))/(r_(1)r_(2)r_(3)) = 4`