SCHWEITZER 2016 Problem 2

Let $K=(V,E)$ be a finite, simple, complete graph. Let $d$ be a positive integer. Let $\phi:E\to \mathbb{R}^d$ be a map from the edge set to Euclidean space, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$ , and the points assigned to the edges of any triangle in $K$ are collinear. Show that the range of $\phi$ is contained in a line.