# The Geometry of the Steiner Tree Problem for up to Five Points

The Geometry of the Steiner Tree Problem for up to Five Points

We would like to find an optimal way of connecting a given finite set of points in the plane, in the sense that the total length of the line segments joining the points is minimal. This optimization problem is referred to as the Steiner tree problem. This Demonstration supports a manual search for such a network of line segments. It can be proved that an optimal network is achieved by adding some points (called Steiner points) to the original set of so-called regular points and finding the minimal spanning tree of the resulting complete graph, where the weights of the edges are the Euclidean distances between the endpoints.