The solution given in Fomin's book is as follows. If such a closed path were possible, then all the line segments could be partitioned into pairs of intersecting segments. But then the number of intersecting segments would have to be even.
How is the first statement true? It seems that we could not have 9 line segments each of which intersects exactly one other line segment in the first place because there will be one left out. I cannot understand the meaning of the first statement.