Inscribe a square in a given circle by following the rules of construction with ruler and compass but... without using the compass.
The center point of the circle is given too.
Name the center of the circle $O$. Draw a diameter $AOB$. Pick another point $C$ on the circle. By Thale's theorem $AC\perp BC$.
Use the method described here to draw lines parallel to $AB$ and $AC$ through $O$. The intersections of these lines with the circle form a square.
We know this is possible
For those that don't know the classic construction with straightedge and compass: Perpendicular through the center, connect the points where the lines and arcs meet.
Poncelet-Steiner shows that straightedge and compass constructions can be done with a straightedge, arbitrary circle and centerpoint:
We can get parallel lines, perpendiculars and transfer lengths. See how here.
Please note, to construct the perpendicular through the center is not directly possible. You need to construct off center through another point first and then transfer it.
Assumption made: Rectangular ruler.
If the ruler is a rectangle, put the corner of the ruler on the center dot,spin around to draw the circle. Then at some arbitrary angle trace along the two sides of the ruler. They are perpindicular, now extend those lines to reach the circle forwards and backwards, and then connect the dots where the lines intersect the circles. That is a square.