Here



B and D become coincidental, as do A and C. From this point we can fold the linkage like this:



And we can swivel the coincided B, D rod around in a circle. Yet another circle of motion happens on the other side, when B and C are made coincidental.

Of these two newly noted circles, one position is in both. Namely where all rods coincide. Therefore, the configuration space should be three circles, attached such that any two intersect at a single point. Try to draw this.