So far, to solve the problem of minimizing a sum of Euclidean norms, it is rewritten as a second-order cone program then solved by interior-point methods. Then its solutions are approximate. In this paper, the problem is treated as a shortest path problem in computational geometry and we introduce an exact algorithm for solving it. The concepts ``final lines" and ``orienting lines" are introduced and the exact solution of the problem is determined by points on orienting lines and a final line. A numerical example is presented.