Splitting a Point from Its Parent

When you select a single midpoint, a point on a path, or a point of intersection, the command becomes Split Midpoint From Segment, Split Intersection From Path Objects, or something similar. If the selected point is a point on path, it’s removed from its path. If the selected point is an intersection point, it’s split from the intersecting objects. In either case, it becomes an independent point and can be dragged anywhere.

Split Point From Circle

Split Intersection From Path Objects