Queen versus rook endgame: to check or not to check? Most tutorials the K+Q versus K+R endgame claim that at some stage, the key is to force the rook away from the king, then it is supposed to be easy: give a series of checks and fork/skewer the rook.
However, when training with computer, it turns out that it's a bit more subtle. For example, I often reach an &quot;almost Philidor's position&quot; below, from which I don't know what to do, so I learned the following sequence:
[FEN &quot;8/8/8/8/8/4KQ2/6rk/8 w - - 0 1&quot;]

1. Qc6 Kh3 2. Kf3 Rg3+ 3. Kf2 Rg5 4. Qe4! Ra5 5. Kf3! Ra3+ 6. Kf4 Kh2 7. Qc2+ Kh3 8. Qc1! Ra4+ 9. Kf3

After black's 4... Ra5, the rook got separated from the king, however, there's no way to win it with checks. Instead, we have to make several quiet moves allowing rook checks, but using the fact that checks run out, and mate threats prevent the rook from rejoining the king. In the final position, Kh2 is the only move that stops both mate threats, but then Qd2 and the rook is finally forked next move. This is all rather non-obvious (to me). A similar situation occurs when the computer suddenly abandons the third-rank defense.
So, my questions are about the position after 4...Ra5, and concretely, about the property that one shouldn't try to win the rook by checks:

is there a rule to recognize such positions, or is this an exceptional setup that one needs to know? The geometry in position after 4...Ra5 does look somewhat special to me, in that white King blocks the e1-h4 diagonal so that the rooks cannot be forked form e1.
if this is a position one has to know, what are other  other positions with a similar property? Among them, are there positions that have no quick win (so that white would need e. g. to allow the queen and the king to rejoin and set up some kind of 3rd or 2nd rank defense)?
For a human, winning Q/R against a computer is more difficult than expected. Indeed, the key is not checking itself but avoiding that the rook checks the king away. Of course, the best concrete strategy in some position may giving some checks first, just to lead the Q to the correct field without losing a tempo. Note that in the final position the Q controls the vital a3 field. Also observe that 1.Qc6 is fastest, but not most lucid in terms of human strategy. I surely had begun with 1.Qh5+ and trying to convert it into the Philidor position. But also here the trick is that attacking the rook hinders it on returning, e.g. : 1.Qh5+ Kg3 2.Qh1 (again, losing a tempo is more elegant, Black to move will be forked quickly) Ra2! 3.Qf3+ Kh2 (otherwise a fork comes soon again) 4.Qd5! (the key move) Rg2 5.Kf3 Rg3+ 6.Kf2 and again, Black is out of checks. (The point being, after your king lands on f3 uncontested, the defense collapses.)