The geometric dilation of three points

Given three points in the plane, we construct the plane geometric network of smallest geometric dilation that connects them. The geometric dilation of a plane network is defined as the maximum dilation (distance along the network divided by Euclidean distance) between any two points on its edges. We show that the optimum network is either a line segment, a Steiner tree, or a curve consisting of two straight edges and a segment of a logarithmic spiral.