By simple graph I mean a graph with no loops or double edges.
If $C$ is a cycle and $e$ is an edge connecting two non adjacent nodes of $C$, then $e$ is called a chord.
I realize that one plan of attack is to choose any node, say $v_0$. Then, since the degree of $v_0$ is $3$ there are $3$ other nodes connected to it. Repeating this argument we will eventually have to reach a node that is connected by an edge to one of the previously used nodes.
I just don't see how this guarantees the existence of a chord.