If I have an MST, and I add any edge to create a cycle, will removing the heaviest edge from that cycle result in an MST?
The important part is that you start with an MST for the original graph. With this extra piece of information, you can build a proof by contradiction, as follows:

Construct the new tree as described (add the edge, check the cycle, remove the edge with the largest weight), now assume for contradiction that this tree is not an MST. This may be for several reasons:


The real MST of the new graph differs in an edge that's not in the cycle. Then the original MST wasn't an MST to begin with, and we have a contradiction.
The real MST differs in an edge that's in the cycle, but then, as the rest of the tree is the same, we can't have removed the heaviest edge from the cycle, and we have a contradiction again.


Therefore the new tree is an MST of the new graph.