Prove that the area of the square is half the square of the diagonal.

The side of the square path will correspond to the variable n.

Consequently, its area, based on the well-known formula, can be represented through n ^ 2.

Now it is necessary to express the diagonal of the square (variable d) through its side, for which we will use the Pythagorean formula known from the school course:

d ^ 2 = n ^ 2 + n ^ 2;

d ^ 2 = 2n ^ 2.

As we remember, S = n ^ 2, we substitute the area in the expression obtained above:

d ^ 2 = 2S.

Then:

d ^ 2 = 2S;

S = d ^ 2/2.

That is, we got the expression that was required in the condition.