Three circles touch each other externally. The tangents at their point of contact meet at a point whose distance from a point of contact is 4. Then, the ratio of their product of radii to the sum of the radii 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`