Circumscribed square
Construct a square with the given 4 points on the sides.
Hint:
Consider two line segments that connect points on opposite sides. Rotate one by π/2. Applying the previous construction, find a line parallel to a side of the required square.
Alternatively, draw a circle with a diameter connecting a pair of points on adjacent sides; it contains a vertex of the square. Repeat this for the remaining two points. Observe that a diagonal of the square passes thru the midpoints of the inner half-circles.