Please help in this difficult math geometry quiz.

 From:  Elang
11167.43 
Hi All,

I think I've solved it, though there are few very tiny errors...



So I made plenty-plenty trial n error by rotating the green circle (on the right) with A2 as its pivot. Every time I rotate this circle, I make a 3pt Circle (red) through A1,A2, and B, to check if this circle intersect all this circle, green circle, and vertical axis in exactly one point... The goal is to make this red circle touch all of A1, A2, B, and C where C must be on the Vertical Axis. It's very impractical but does the job, I guess..

As you can see, there is very tiny difference at the red circle radius where the vertical radius has a 0.0000001 deviation from the other. Also, the angle of C and B differs about 0.0000011 deg. I think this is good enough (?)

I'm still looking forward for much simpler and better solution, though... so please write anything should you feel necessary.

Cheers!