If sum of two numbers equals with sum of two other numbers , then their product will never be equal . If a+b = c+d ,then ab ≠cd The viceversa is also true.if ab = cd then a+b ≠c+d.

Let L and B be the length and breadth of a rectangle. Then its perimeter become 2(L+B) = P and its area A = LB.Another rectangle with same  P  has length L’ – (L-a)and breadth B’= (B+a). To have same area A’ = L’B’ =(L-a)(B+a) = LB +a(L-B) – a² = LB. Or (L-B) = a which demands  L = B+a and B = L-a where the length and breadth are interchanged.. 

P = 2L + 2B.  and area A = LB.When P remains same B = (P/2) – L, the ares of the equi-perimeter rectangles A = L[(P/2) – L] = {PL/2) – L^2. .The chnnge in area due to the change in the dimension of the rectangle having same perimeter dA/dL = (P/2) – 2L A is maximum when dA/dL = 0 or P = 4L which demands L =B  

Let L be the side of a square. Then its area A = L² and its perimeter P = 4L. When it is reshaped into a rectangle with length L’ = L+x and breadth B” = L-x. Then the area of the rectangle A’ = L’B’ =(L+x)(L-x)= L² – x² = L² or x = 0..The area of a rectarangle will be maximum when it becomes a square with same perimeter.