Shortest path joining three points

If the triangle ABCABC has an angle, say at AA, greater than 120° then the shortest path is BB to AA to CC. If all the angles are less than 120° then the sum of the line segments AOAO, BOBO and COCO is the shortest path (See figure). Torricelli showed that OO 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 OO. Simpson showed that the line sgments joining each vertex of the triangle to the opposite vertex of the equilateral triangle intersect at OO. The lengths of each of these line segments are equal to the minimal path length AO+BO+COAO + BO + CO. Fasbender showed that this minimal path length was equal to the maximum of the latitudes of all equilateral tringes circmscribed about ABCABC.