Consider a square of perimeter 4L. Next, construct a rectangle with its lower left corner coinciding with the square’s. The perimeter of the rectangle is also 4L, with base L + x, 0 < x < L. Using only geometry, prove that the area of the rectangle is less than the area of the square for all 0 < x < L, hence conclude that the square yields the largest area among all rectangular shapes with the same perimeter as the square. Note: No fair using any algebra for solving this problem. Remember, algebra was developed centuries after Euclid passed!