Answer:
AB = (2+2√3)r
Explanation:
All three sides of an equilateral triangle equals 60° each.
Given that the circles are equal and are inscribed in a triangle, the angle bisectors pass right through the center of the circle present in front of that angle.
For example a figure has been attached with the answer, where angle bisectors make a triangle with center of the circle and a perpendicular projection of the center on side AB.
Finding AB:
Let us divide the side AB into three parts. One is the line joining the center of the two circles which is = 2
Then we have two equal parts, each joining one vertices with the center of the circle.
Let us assume that there is a point P on the side AB which forms a line segment PO₁ ⊥ AB.
We have the right angled triangle APO₁. Angle A = 30° PO₁ = r
let the base AP = x
We know that tan 30° = perp/base
1/√3 = r/x
=> x = √3 r
Hence Side AB = √3 r + 2r + √3 r
AB = (2+2√3)r