BMO1 Geometry Question

This is question 4 from BMO1 2014 and I've gotten stuck on it. I've drawn the diagram:

All it tells you is that the minor arcs AF and BF are equal and it wants you to prove that the lines AB and PQ are parallel. What I've done so far is in the attachments, I can write all of the red angles in terms of A,B,C,D and I know that A+B+C+D = 180 deg.
There's lots of similar triangles all over the place but they're really not helpful. I've thought about doing a converse corresponding angle proof with the green angles, but for that I need to know PQF. Any help would be appreciated

Let ABCD be a cyclic quadrilateral where F is the midpoint of the arc AB of it circumcircle. Let AC and FD intercept at P and BD and FC intersect at Q. Prove tha PQ and AB are parallel

I've had a look at it for quite a while and can't see it. If you need the solution urgently, your best option is probably to pay £15 for the UKMT 2014-15 yearbook (available at http://shop.ukmt.org.uk/ukmt-books/2014-2015-yearbook) which will include the BMO solutions. Alternatively, if you're willing to wait, you can make a Freedom of Information Act request to the University of Leeds, which the UKMT is a part of.

I think the solution would go something like this:


Might be wrong, but I think that's how I did it

I think the solution would go something like this:


Might be wrong, but I think that's how I did it