11It'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.
a circle is inscribed in a equilateral triangle of side a.find the area of any square inscribed in the circle.
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39now ... 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}