Geometry

Let $s$ be the square's side length, and $s\sqrt{2}$ the length of its diagonal. Hence, the sum of its sides and diagonals $N$ is $4s + 2s\sqrt{2}$ . Once you solve for $s$ the square becomes easily constructible.

$\displaystyle s = \frac{N}{4 + 2\sqrt{2}}$

The $\color{#69047E}{CROSS}$ divides the given line segment in the ratio $\sqrt 2 : 1$ . So, we can construct our square.

Since all squares are similar, we begin by constructing a square arbitrarily. Let a' , d' denotes its sides and diagonals respectively and let a, d be the corresponding elements of the given square. From the similitude of the two figures $\frac{a}{a'}$ = $\frac{d}{d'}$ => $\frac{d}{a}$ = $\frac{d'}{a'}$ By componendo $\frac{a+d}{a}$ = $\frac{a'+d'}{a'}$ => $\frac{a+d}{a'+d'}$ = $\frac{a}{a'}$ In the last proportion we know 3 terms for a+d is given . Hence, the segment a may be constructed as a fourth proportional, and the problem is reduced to constructing a square given its sides.