Like the other answers, the answer is:
However, a correct explanation is more complicated. In particular:
The second support plane needs to do two out-and-back flights on each side of the planet. Here's a full breakdown (important new steps bolded):
There are three planes (AE, S1, and S2). They each start with a full tank.
1. They all fly 1/8th of the way.
2. AE, S1, and S2 all now have 3/4 tanks. S2 transfers 1/4 tank to S1 and turns back. AE now has 3/4 tank, S1 has a full tank, and S2 has 1/2 tank.
3. S1 and AE continue for another 1/8th of the way.
4. S1 has 3/4 tank while AE has 1/2 tank. Meanwhile S2 returns home with 1/4 tank to spare.
5. S1 transfers 1/2 tank to AE and turns back. S1 now has 1/4 tank while AE has a full tank. S2, refueled to a full tank, sets off again.
6. All planes fly another eighth of the way. AE is now 3/8ths of the way with 3/4 fuel, S1 and S2 rendezvous 1/8th of the way (S1 has just run out of fuel and S2 has 3/4 tank).
7. S2 transfers 1/4 tank to S1 and turns around. S2 now has 1/2 tank and S1 now has 1/4 tank.
8. The planes fly another eighth of the way: S1 arrives at home with an empty tank, S2 arrives at home with 1/4 tank. AE reaches the 4/8ths mark with 1/2 tank.
9. S1 and S2 refuel and set off again to do the whole thing in reverse on the other side of the planet.
[Edit: after discussion on the other answers, it was noted that:
If S2 gives 1/4 tank to both S1 and AE (thereby transferring 1/2 tank), then it also works (and S2 only needs to make one trip).
The phrasing of both other answers led to my confusion on this; they are both correct under this clarification. My answer is then just an alternate approach.]