First, the two nodes at the bottom have to be connected to each other. Then they both must connect to the 8 and the 6. This is due to rule 2. Without a vertex in the center, this makes a path of length three leading to the 6, so another path of length three needs to connect to the 6 as well (or a combination of paths that sum to length three), because of rule 3.
It can't connect to either of the two nodes on the right, because that length is only two. The only two nodes of distance three are the 6 in the center of the puzzle and the O next to the 8 in the bottom left. To get to either of those nodes, the segment path from the 6 has to pass through the center vertex of the bottom hexagon.
Now the 8 on the left still needs segments that sum to length five to connect to it. The only paths available are to the O node next to it (distance one) and the vertex at the center of the hexagon. You can see that no matter what, it must connect to the vertex in the center - but that vertex must also connect to the 6 on the right, which is already connected to the 8 along the bottom. This creates a loop, which violates rule 4.