Consider $\displaystyle n$ points randomly distributed in space. The goal is to find the graph that has the smallest Euclidean length that connects these points. The answer is clearly the minimum spanning tree $\displaystyle G_0$ which has $\displaystyle n-1$ edges. Now suppose we are required to use exactly $\displaystyle n$ edges and find the smallest length graph $\displaystyle G_1$. Clearly, $\displaystyle G_1$ will contain exactly one cycle. Now suppose we remove the longest edge from this cycle and obtain graph $\displaystyle G_1'$. I am wondering if $\displaystyle G_1' = G_0$ for any configuration of points. My intuition is that it is true, and I'm not able to find a counter example. However, I am not able to prove it. Can someone help?

Thanks.