I have a rectangular table. Can I cut it into 3 pieces to create a removable center leaf?
It is possible. Making good cuts is going to be the trickiest part.

If you have a large table saw that can accommodate the table top that will give you the best results. Otherwise you can use a straight edge with a sharp, fine toothed blade in a circular saw.  Use painter's tape over where you are going to cut. Mark the line then score the line a couple of times with a sharp utility knife to reduce chip out. If you can rent or borrow a good track saw that would be better at making a straight cut and minimizing tear out.

Next you'll need to attach hardware the pieces. Check out Woodworkers Hardware http://www.wwhardware.com/fasteners-table-hardware/table-hardware/table I've gotten a few things from there and they have good prices and variety. I would use a combination of the leaf alignment pins to align the leaf and the table slides.