Getting rotation angles using 3 known points

I've got 3 known points in 3d space. the points are mounted on a rigid construction so their relative position stays the same.

This little system of 3 points can move and rotate freely in 3d space.

Now at any given moment I know these 3 points (x,y,z).

How do I calculate what rotation angle you would need (over the fixed x,y,z system) to get in that position?

There's a lot of info available how to rotate objects in 3d over specified angles (mostly using quaternions), but I need to do it in reverse so to say. I know the final position but not the rotations to get there.