Question 860521
If the apothem is exactly half of the side length, then the apothem, which is the center to the midpoint of the side, is exactly the distance from said midpoint to the nearest corner. Therefore, the triangle formed by the center, midpoint, and corner is an isosceles right, with a 45° at the corner. Two 45°s make a right angle, and so the figure is a square.
 
{{{drawing( 300, 300, 0, 6, 0, 6, line(0,0,0,6), line(0,0,6,0), line(0,6,6,6), line(6,6,6,0),

line(3,0,3,3),line(3,3,6,0),
locate(2.5,1.5,3), locate(4.5, 0.5, 3), locate(5.25, 0.35, 45))}}}