What makes you say “I know I have to use radical axis”?
Personally I'd treat this as a special case of Pascal's Theorem, if you can build on that:
Pascal's theorem states that given all the other incidences, the three points $M,N,P$ will be collinear in the above situation. Now you can imagine moving $A_1$ and $A_2$ closer together. In the limit, when they are the same point, the line connecting them will have become a tangent. Likewise for the other two. So all the dark blue lines would become tangents, all the cyan ones are opposite connecting lines. It doesn't even have to be a circle, any conic will do.