I plotted 4 random points on the 2D plane and calculated the shortest and the longest paths joining all 4 points together. The image below shows the longest (blue) and the shortest (black) paths that connects all 4 points in a continuous line. There are totally 6 line segments connecting the 6 possible pairs of points. Only 3 line segments are needed to connect all 4 points. My simulation shows that there are no overlapping line segments for the two paths. Similar results are obtained every time.
Is there any prove to this and has anyone heard of it?
If some of the lengths are equal (e.g., if the points are vertices of a square), it may happen that a shortest path and a longest path have an edge in common. However, whenever the shortest and longest path are unique, your observation holds.
Note that for any three-edge path running through all four vertices, the complement (=the other three edges) form a path (and not a closed triangle and not a Y-shaped tree). The sum of the length of such a path and of the length of its complement path is of course constant. Thus, there cannot be a path that is longer than the complement of the shortest path, because the complement of that longer path would be shorter than the shortest path.