Using a compass one should be able to find the intersect at pt X. The way I did to get the arc (C). X wouldn't be a value but...
Friend, I like puzzles well enough, but I cannot even figure out what you are trying to figure out.
What are the constraints for where the "next hole" is supposed to be, with respect to the location of the existing hole?
Is it something like this?
We had an order to make 20 or so circle signs. Printed the vinyl and applied to the circles. "Oh, now we need two holes drilled so they can be attached to U-Channel posts". Hmmm, they all have a logo at the top center, how about we put the first hole there and then figure 12" down. But we need to ensure the sign isn't crooked so we need to find at what point on the arc the hole should be.
So, is the question, "How do make a vertical line that intersects hole A?"
You could draw two arcs of equal radius, then use their intersection to get the vertical line from hole A.
Yes, exactly, but at what radius?
For those two arcs I just showed? Use whatever radius is convenient to put their intersection point fairly close to the bottom of the circle. If both arcs have the same radius, then their intersection will lie on the vertical line.
I'm only using those to get the vertical line, not the "12 inches below hole A" part of it.
hang the circle from a nail on the wall then after it settles swinging, put a level vertically from the nail down passed where your next hole should be. Adjust level as need then mark the point on the arc at 12".
Sign will be crooked anyway after they pound the U-channel post in. lol
If you just wanted to Git 'Er Done, then my first question would have been, "Is there anything on the sign that you can use to eyeball horizontal/vertical well enough?".