Solution:

There are only three such configurations of three non-crossing diagonals, as shown in the diagram

In the first, three diagonals form a broken polygonal line. The starting point can be any of the six vertices. In the second, the three diagonals all converge in a single vertex, and this can be any of the six. In the third, the three diagonals form a triangle joining alternate vertices. This yields only two ways. Hence the total number of ways is 6 + 6 + 2 = 14