Area of paralelogram $BEDZ$ is equal to $DE*1=DE$. On the other hand same area is equal to $EB*AB$. Area of whole square is $AB^2$ and Is also equal to sum of area of triangles $EAD, ZCB$ and parallelogram $BEDZ$ so we have $AB^2=3AB*EB$. Since $DE=EB*AB$ we have $DE=\frac{AB^2}{3}$. From triangle $EAD$ we have that $DE=\frac{AB\sqrt{13}}{3}$, so from last two equalities we get $AB=\sqrt{13}$ and so the area of square is $13$.