Then draw a circle tangent to the new circle and the original two. Continue building a chain of circles each touching the two given circles and its predecessor in the chain. It may happen that, for some n, the n-th circle will touch the first circle in the chain. Then this will happen regardless of the position of the starting point. T transforms the two original circles into concentric circles.
T preserves tangency.
For concentric circles, i.e., circles with the same center, the statement is quite obvious. The radii of the given circles and n must stand in a certain relationship for the chain to close on itself. Whether it does or not does not depend on the position of the starting point. Two polygons with the same number of vertices inscribed into a fixed circle. Vertices of the polygons may be moved along the circle.