Reduction from Vertex Cover
This is known as the $k$-center problem or the planar minmax Euclidean facility location problem.  It is a special case of the minimax facility location problem for the Euclidean metric.

The $k$-center problem is known to be NP-hard.  Wikipedia has references to the literature, where you should be able to find an explicit reduction.

The special case where $k=1$ can be solved in polynomial time.