I think everyone is getting tied in knots a little here.
I am an Architect . . here are the facts : -
1 - Think of the towers as tall square 'tubes'. The external skin IS structural and acts with a very lightweight lattice floor plate structure spanning to the central core (to acheive collumn free commercial spaces).
2 - It was not 'big steel girders' which 'melted'. The towers were the worlds first masonry-free high rise towers and the steel was protected with lightweight fire-board (drywall as I think you guys call it over there).
3 The aircraft hit the buildings at slightly different angles. One had more damage to the central core than the other and in both cases the impact and explosion 'blasted off' varying amounts of fire protective boards in each case.
4 It is widely accepted that it was failure of the connection of the lattice floors to the central core which precipitated the collapse. It doesnt take much heat to warp lightweight lattice steel, especially when its fire protection is gone, plus the fact that the external stressed skin facade was seriously weakened in the initial impact.
5 When the critical number of lattice connections had failed in fires which were burning entirely uncontrolled (fueled not only by aviation fuel but by offices full of carpet furniture and paper) then the floor plate would collapse. When that happened, the weight of the concrete slab on top of the lattice structure would fall 12' and slam into the floor below and so on and so on.
If you are a construction professional its not so hard to understand. Remember, these were the days before progressive collapse building codes were introduced.
http://www.civil.usyd.edu.au/wtc.shtml
The aircraft with no windows? . . .well thats maybe another story.