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.