I can't solve this problem in any way (marked "trouble" on gif)
"A" and "C" are separate IK-chains that work well. But, I cannot connect them correctly.
"C.2" bone locked rotation, only location transform. In this example was try to use Limit distance constraint from "B.1" bone to "A.IK-target" bone. And "C.IK-target" bone use Locked track constraint to "B.1" bone.
Update:
This is rig of door. All joints must be connected and must not separate in animation.