Question

1.prove the radical axis of a pair of circles is the common tangent line. help

Answer 1

Well, the only situation where the radical axis becomes a common tangent is when the circles are tangent to each other, meaning they only share one common point.
To prove that, I would define two circles that are tangent to each other, intersect them and then prove that the axis will be perpendicular to the line containing both centers.

Answer 2

Well, try with something like this, in regards to Euclidean geometry, we can make two circumferences with different radiuses be tangent at some point A.


We weill choose an arbitrary point on the line that is tangent, in this case "B", if you can prove that the line AB is perpendicular to line CC' you'll have proven that the line is indeed the radical axis, since its a line that goes through the point of tangency of the circumferences.
And since it is te only point that the two circumferences share, it must be a tangent, thus proving that in this specific context, the tangent and the radical axis are the same line.

Answer 3

Something that would be useful, as a side theorem to prove is the statement: "Any tangent to a circle will be perpendicular to the radius proyected from the point of tangency to the center".
meaning that:

if you manage to prove that the line "t" is perpendicular to the segment CP, you'll have proven that the radius to a point will always be perpendicular to the tangent line.
As a tip, I suggest you look at the definition of "distance".