Figure description: An equilateral triangle encloses the two circles touching each other externally. Two sides act as common tangent to both the circles, whereas the third side is tangent to the larger circle only.
What will be the ratio of the perimeter of the smaller circle to that of the equilateral triangle in the given figure?

The bigger circle is the incircle of the triangle ABC. We know that incenter of an equilateral triangle is also it's centroid. It means that OD = 1/3rd of AD.
It also implies that OE = OD = AE = 1/3rd of AD.
AE is the median for the smaller equilateral triangle and thus radius of the smaller circle = 1/3rd of AE ==> $$\frac{1}{3}\times\ \frac{1}{3}\times\ AD=\frac{1}{9}\times\ AD$$.
AD = $$\frac{\sqrt{\ 3}}{2}AB$$. This gives the smaller radius = $$\frac{1}{9}\times\ \frac{\sqrt{\ 3}}{2}AB=\frac{\sqrt{\ 3}}{18}\times\ AB$$ and Circumference = $$2\pi\ \times\ \frac{\sqrt{\ 3}}{18}\times\ AB\ =\ \frac{\sqrt{\ 3}\pi\ }{9}\times\ AB$$.
And the perimeter of the triangle = 3AB.
Thus, the required ratio = $$\frac{\sqrt{\ 3}\pi\ }{9}\times\ AB\ \ :\ 3\times\ AB=\ \pi\ :\ 9\sqrt{\ 3}$$.