Based on The square and the compass
The rules are almost the same. (The only difference is the actual task, marked in bold.)
You have a compass and a pencil but no scale/straightedge.
Your job is to mark four points on a plane paper that would form a square if joined. You are given two points on a plane paper. Your job is to find the midpoint of line segment formed by the two points, if joined. Your result has to be perfectly accurate (not approximate) and should be possible in a finite number of moves. The following moves are valid.
- Make the compass radius equal to the distance between two already marked points.
- Draw a circle with any marked point as centre.
- Use any intersection of arcs/circles as a marked point.
- Select a continuous region (either an arc or a 2D region) and mark an approximate point. For example, I could draw an arc and then mark a point that is approximately (but not exactly) at the centre.
I do not know how to solve this.
I do know how to do it in infinite moves, though.