Something I have only come to realise recently, but which may have been obvious to many, is that when a boat is on a reach, the optimal angle of the sail to the wind is always the same and it is only the angle of the boat to the sail that alters:
Because the sail is acting like a wing when on a reach, the "lift", or propulsive force from the wind is at 90 degrees to the sail.
As you can see from the diagram, when the boat is on a close reach, it approaches 90 degrees to the propulsive force of the sail, at which point it stops moving forward.
On the other hand, when the boat is between a beam reach and a broad reach, the hull of the boat is travelling more or less in the same direction as the propulsive force, which is why that is the fastest point of sail.
This is consistent with JollyGreen's diagram, which suggests that if the optimal angle of a sail to the wind is 45 degrees, then the fastest point of sail is at a right angle to that - 135 degrees.
I drew the above diagram, so don't assume it came from an authority on sailing.