Another Proof of the Steiner Result for Three Equidistant Points in Euclidean Space and an Analogous Proof in Hyperbolic Space

In the early 19th century, Jacob Steiner wanted to find the shortest path to connect three villages. He concluded that the shortest path depended on the angles of the triangle created. If all the angles were less than 120 then the shortest path involved a fourth interior point, a Steiner point, at which the segments from the vertices all meet at 120°. Our research group developed a new slicing method that can be used to prove which paths are minimal. To demonstrate this new method, we first give a new proof of a particular Steiner result. Then we use it to prove an analogous result in hyperbolic space; that is, the shortest path between three equidistant points in hyperbolic space is formed by hyperbolic geodesics that meet at 120°