KöMaL 2016 A683

Let $K=(V,E)$ be a finite, simple, complete graph. Let $\phi\colon E\to\mathbb{R}^2$ be a map from the edge set to the plane, 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 are collinear. Show that the range of $\phi$ is contained in a line.