Three circle touch one another externally. The tangents at their points of contact meet at a point whose distance from the point of contant is 4. If the ratio of the product of the radii to the sum of the radii of the circle is `lambda,` then `lambda/2` is .........

Explanation

Let us consider three circles with centers at A,B, and C with radii `r_(1), r_(2) and r_(3)`, respectively, which touch each other externally at P,Q, and R. Let the common tangents at P,Q and R meet each other at O. Then
`OP = OQ = OR = 4` (given) (length of tangents from a point to a circle are equal)
Also, `OP bot AB, OQ bot AC, OR bot BC`
Therefore, O is the incenter of `Delta ABC`

Thus, for `Delta ABC`,
`s = ((r_(1) + r_(2)) + (r_(2) + r_(3)) + (r_(3) + r_(1)))/(2)`
`= r_(1) + r_(2) + r_(3)`
`rArr Delta = sqrt((r_(1) + r_(2) + r_(3)) .r_(1).r_(2).r_(3))` (Hence s formula)
Now, `r = (Delta)/(s)`
or `4 = (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)))/(sqrt(r_(1) + r_(2) + r_(3)))`
or `(r_(1) r_(2) r_(3))/(r_(1) + r_(2) + r_(3)) = (16)/(1)`
or `r_(1) r_(2) r_(3) : (r_(1) + r_(2) + r_(3)) = 16 : 1`