Is the path between 2 vertices of a Minimum weight tree of a graph the shortest path between those 2 vertices? Suppose we have an undirected, connected graph, $G_1$
If you have a minimum weight spanning tree $G_2$ for graph $G_1$. Is it possible to find two vertices in $G_1$ which is has a shortest path that is not present in $G_2$?
Intuitively it seems that this is not possible otherwise that shortest path would be part of $G_2$ but I would like to know if I am wrong 

HINT:
Think of a triangle $ABC$ with $AB=3$, $BC=4$, $AC=5$. The minimum spanning tree contains the edges $AB$, $BC$ and $AB+ BC > AC$. 
