Hi.

I have used : The inradius of an equilateral triangle has radius r = a/3

where a is the altitude

After making a new tangent to the incircle parralel to the AB,

we get a new triangle similar to the earlier triangle

( ie; an equilateral triangle with side S'= S/3)

the incircle of this triangle is our next required circle

and thus we get its radius(r) as a'/3

Hence the new radius becomes

Similarly

Thus the sum now becomes

This in an infinite GP with first term

And common difference

Thus the sum of terms is given by

We needed to get it for three sides so sum is