What's the idea behind constructing the radical axis?

Question

I know how to construct it, but I don't understand why every point in the radical axis has the same power of a point with respect of the two circumferences.

Answer 1

That is by a definition. Radical axis is a set of points that have equal power to both circles.

Now you have to prove that radical axis is actually a line.

Answer 2

Suppose $K$ has the same power with both circles: $$ KB^2-r^2=KV^2-R^2. $$ Then any other point $J$ on the perpendicular at $K$ also has the same power, because by Pythagoras theorem we have: $$ JB^2-r^2=JK^2+KB^2-r^2=JK^2+KV^2-R^2=JV^2-R^2. $$