Shortest path joining three points
If the triangle has an angle, say at , greater than 120° then the shortest path is to to . If all the angles are less than 120° then the sum of the line segments , and is the shortest path (See figure). Torricelli showed that is the point of intersection of the circumcentres of the equilateral triangles drawn on the three sides. Cavalieri showed that the line segments of the paths form angles of 120° at . Simpson showed that the line sgments joining each vertex of the triangle to the opposite vertex of the equilateral triangle intersect at . The lengths of each of these line segments are equal to the minimal path length . Fasbender showed that this minimal path length was equal to the maximum of the latitudes of all equilateral tringes circmscribed about .