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Let the sides of the right triangle be a and b
Therefore, the hypotenuse is root(a^2 + b^2)

Let the side of the square be x
Therefore, the diagonal of the square is x root2

The area of the triangle, considering a and b is 1/2(a*b).
The area of the triangle is 1/2(root(a^2 + b^2))(x root2) considering the hypotenuse and its corresponding height, the diagonal of the square.

Therefore, x=(ab)/(root 2)(root a^2 +b^2) or (ab)(root 2a^2 - 2b^2)/root(4a^2 -4b^2)