I think its eaasy...bt cudnt get the ans.

It's better with trigonometry.
Inradius of the triangle is known.....now the side of the square be x, so 2x²=4r² so x²=2r².....where r is the inradius of the triangle.

now ... r=\frac{a}{2\sqrt{3}}

and for any square inscribedin the circle ... we have the diagonal is twice the radius of the circle..

let the side of square be b

then \sqrt{2}b=\frac{a}{\sqrt{3}}\Rightarrow b=\frac{a}{\sqrt{6}}

so area of square =b^{2}=\frac{a^{2}}{6}

thank u everyone......