Considering a medium creature, with 5ft of reach, wielding a reach weapon such as a longspear.
By the rules:
With a typical reach weapon, you can strike opponents 10 feet away, but you can't strike adjacent foes (those within 5 feet).
Squares within 5 feet would be all adjacent squares to the creature, including diagonal ones (first diagonal being counted at 5ft).
It appears clear that the weapon will allow attacks into four squares: those 5ft away from the creature in the north, east, south and west directions. They are 2 squares away in a straight line, thus 10ft away, within reach.
But now, what about the squares diagonally away, beyond 5ft, north-east, south-east, south-west and north-west of the creature? According to the rules, the second square of a diagonal line is considered 15ft away when it comes to movement. Does this also apply to attack distances?
Also, what about squares reachable by going diagonally once, then straight up/down/left/right? This would usually count at 10ft of movement, so are they attackable?
Here's a diagram:
\begin{array}{c|c}
\phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} \\ \hline
& ? & ? & X & ? & ? & \\ \hline
& ? & & & & ? & \\ \hline
& X & & C & & X & \\ \hline
& ? & & & & ? & \\ \hline
& ? & ? & X & ? & ? & \\ \hline
\\
\end{array}
\$C\$ is the creature, blank spaces are where I assume you cannot attack, \$X\$s are where I assume you can attack and \$?\$s are where I'm not sure you can.
In searching for answers to this, I've come across an interesting point on the Paizo forums. If you are in one of the far corners and you consider these not threatened, then you can diagonally move towards the character and never provoke an AOO (that is, per RAW)… and yet, logically, a threatened area should make an uninterrupted circle around the creature. This may explain why 3.5 made an exception out of this.
Best Answer
Yes, that is precisely why 3.5e made that exception, and it’s also why Paizo issued an official FAQ that changed Pathfinder’s rules to add the same exception, as @caps reports in this fine answer that you should go upvote.
Thus, a reach weapon can attack the following \$X\$’s from \$C\$:
\begin{array}{c|c} \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} \\ \hline & X & X & X & X & X & \\ \hline & X & & & & X & \\ \hline & X & & C & & X & \\ \hline & X & & & & X & \\ \hline & X & X & X & X & X & \\ \hline \\ \end{array}
Before the FAQ change
However, the FAQ entry that caps reports did not exist at the time this question was asked. At that time, you did not get the four corners, and so could only attack these \$X\$’s from \$C\$:
\begin{array}{c|c} \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} \\ \hline & A & X & X & X & & \\ \hline & X & B & & & X & \\ \hline & X & & C & & X & \\ \hline & X & & & & X & \\ \hline & & X & X & X & & \\ \hline \\ \end{array}
Here, it would appear that someone could step from \$A\$ to \$B\$ to avoid an attack of opportunity altogether. However, even before the FAQ just changed this to be like 3.5e, the developers at Paizo had ...for lack of a better word, we’ll call it a clarification, though it honestly just confused me more. From an earlier FAQ:
Basically, the idea was, under the rules at the time, you didn’t threaten 15 ft. away, so you don’t get the corner, but you did threaten 10 ft. away and there’s no way to move from 15 ft. away to 5 ft. away without passing through a point that is 10 ft. away. Thus, someone moving from 15 ft. away on the diagonal to 5 ft. away on the same diagonal was going to provoke even under these rules.
So the enemy at \$A\$ moving to the point marked \$B\$ towards \$C\$ with a reach weapon provoked an attack of opportunity (assuming this isn’t a 5 ft. step of course), because somewhere between \$A\$ and \$B\$, there is a point that is 10 ft. away from \$C\$ that the enemy has to pass through.
Presumably, you would have adjudicated the enemy’s position for the purposes of the attack of opportunity as being \$A\$, though this was never made clear. In this sense, the end result was identical to the 3.5e/post-FAQ version for movement towards you: creatures leaving that corner square to enter a square inside your reach provoked an attack of opportunity as if you threatened that square. You were not eligible to make an attack of opportunity if the enemy performs any other action that provokes from \$A\$, including movement in other directions, because you do not actually threaten it.
This was a headache. Even before the FAQ changed things to match 3.5e, that was precisely what I recommended: