The sharp circle square
Let there be a square \(abcd\) with a side length of 1. For each side, mark its midpoint and cast a ray outward perpendicular to the side. Mark a point on each ray with distance \(n\) from its respective midpoint, and pass a circle through the points of the side and the point we just marked. Find an \(n\), so that the center of circle is on the midpoint of the opposite side of the square.