If the flag is enlarged by scale factor $k$ and then by scale factor $\frac{1}{k}$, how large will it be at the end? Will it have changed the way it is facing?

You might find it useful to use vectors for the proof at the end of this question. For example, you could write $\mathbf{x}$ for the vector from the first centre of enlargement to the second centre of enlargement, and $\mathbf{a}$ for the vector from the first centre of enlargement to the foot of the flagpole. If you can prove your answer for the foot of the flagpole, do you need to do any extra work to prove it for the rest of the flag?