I have finally conquered the puzzle. The solution looks something like
Every piece now has an A point, a B point and a C point. I was just about to rule out 5 as a point when the last possible combination of pieces just worked!
Assuming I've already ruled out the two other points of 5 as being the top, then:
- 5 forces 7 at its bottom.
- On the LHS of 7 I must place either 8 or 9.
- On the bottom of 9 I must put either 2 or 6.
- 2 forces 3 on its LHS.
- 7 forces 6.
- 6 forces 4 and then this solution doesn't work.
- Going back to follow on from step 2, on the RHS of 7 I must put either 2 or 6.
- If I have a 2 I can place pieces below the 2 and the 8 but then can't place a piece between these.
- Going back to follow on from step 7, 6 forces 1.
- 9 forces 3.
- This solution works!