A circle is inscribed in a equilateral triangle of side a. Find the area of the square inscribed in a circle.

let r be the radius of the circle. after drawing the necessary diagram, one can see that r/(a/2) =tan30^o, r=a/(2√3)

the diagonal of the square is 2r, therefore side length is r√2, therefore area of square is a²/6 

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take equilateral triangle in such a way that its vertex lies at (a/2 , 0 ) , ( -a/2,0) and (0, √3a/2) . then the centre of the incircle lies at  ( 0, a/2√3 ) so the radius of the incircle is also a/2√3  which is half the length of the diagonal of the square lying inside the circle now on solving side comes out to be a/√6  . therefore area of the square is  a²/6 .