ABC is an equilateral triangle while PQRS is a rectangle, then what is the area of PQRS if each side of the $$\triangle$$ABC = 10. The side of the rectangle passes through the center O of the circle?

The centre of the circle would act as the centroid of the triangle, from where all the median shall pass. 

The length of the median is $$\ \frac{\ \sqrt{\ 3}}{2}\times\ 10$$, which is $$5\sqrt{\ 3}$$ cm. 

Now, this median shall be divided in 2:1 by the centroid (centre of the circle), and hence the smaller part which is the radius of that circle becomes $$\ \frac{\ 5}{\sqrt{\ 3}}$$cm (the breadth of the rectangle).

Double of this shall be length of the triangle, which is $$\ \frac{\ 10}{\sqrt{\ 3}}$$cm.

Hence, the area of the traingle is $$\ \ \frac{\ 50}{3}$$cm, or 16.67 cm.