The middle sized square passes through the centres of the four circles. Each side of the middle sized square together with the edges of the outer square creates a right angled isosceles triangle with angles of 45$^\circ$. Thus the angles these sides make with the inner square are also 45$^\circ$.
Each side of the middle sized square bisects the area of the circle. The inner half of that circle is made up of two shaded segments with angles of 45$^\circ$ which together are equal in area to the unshaded right angled segment.