Answer:
AB = (2 +2√3)r
Step-by-step explanation:
Let X be the point of tangency of circle O3 and AB. Then length XO3 is r. The triangle BXO3 is a 30°-60°-90° right triangle. You know this because BO3 bisects the 60° angle at B of the equilateral triangle ABC.
A 30°-60°-90° triangle has side lengths in the ratios 1 : √3 : 2. That means side XB of triangle BXO3 has length r√3. The distance from A to the point of tangency of AB with circle O1 has the same measure.
Of course the distance between those points of tangency is the same measure as the distance between centers O3 and O1: 2r. So, the total length of AB is ...
AB = r√3 + 2r + r√3
AB = (2 +2√3)r
