I'll give this a go, there must be lots of ways of doing it and mine won't be the most elegant. I'll attempt to find the centre of the arc.

Let's first define B', C', D', G' to be the points that you have marked (plus the centre of the circle) but from a set of coordinate axes that are more convenient. I choose to place the point B' at (0,0,0). To do this we use:

B'=B-B=(0,0,0)

C'=C-B

D'=D-B

G'=G-B

Now let's define C" as some point on the line from (0,0,0) to C'. We can find a candidate for C" using

Now the angle between DB and BG is given by:

Consider the triangle C'G'B' this has the internal angle so the distance from B' to C' is given by:

And finally the centre of the arc is located at:

or