We are given a rectangular box (a PRISM) with dimensions "a", "b" and "c".
    The surface area of the box is 288 sq. cm.
    The 3D diagonal of the box is 12 cm long.
    Prove that the box is a cube.

    Let us consider the quadratic form 

        Q(a,b,c) = .     (1)


     We have Q(a,b,c) =  +  +  = 

                      =  - .   (2)


     As everybody knows  (or MUST know),

         is the square of the length of the 3D diagonal, so it is equal to  cm^2.

     Therefore the term    in (2) is equal to   = 288 cm^2.



     Again, as everybody knows (or MUST know),

          is the surface area of the rectangular prism.

     Therefore,  the term    in  (2)  has the given value of 288 cm^2 under the given condition.



     Now it is clear that at given data the form  Q(a,b,c)  is equal to zero.

     But  Q(a,b,c) is the sum of squares, and can be equal to zero if and only if a = b,  b = c  and a = c.



     It implies that  a = b = c  and the prism is a cube under given condition.