Given a set of points $S=\{p_1\dots p_n\}$ of points in the plane, we define the cost of a pair $w(p_i,p_j)$ as the square of the geometric distance between them
$$w(p_i,p_j)=\| p_i-p_j\|^2.$$
A path from $p_1$ to $p_n$ is a sequence of points from $S$ (e.g. $(p_1,p_4,p_{20}, p_3, p_n)$. The cost of the path is the sum of the cost of each individual edge. We seek the cheapest such path.
Show that any edge that is not in the Delaunay Triangulation $DT(S)$ will not be in the shortest path from one point of $S$ to another.