jmc

From each vertex $V$, we can draw 3 diagonals: one to each vertex that is not $V$ and does not share an edge with $V$. There are 6 vertices in a hexagon, so we might be tempted to say the answer is $6\times 3=18$. However, note that this counts each diagonal twice, one time for each vertex. Hence there are $\frac{18}{2}=\boxed{9}$ distinct diagonals in a convex hexagon.