If the closest point meets the range requirement, but if it is out of arc then the shot is taken out of arc and doesn't fulfill Tacticians triggering condition.
And this is completely wrong. A ship is still within arc even if the closest point is outside of its arc. Read page 10, where being within arc is defined by ANY part of the ship's base being within the arc area. You are only taking an out-of-arc shot if no part of the ship's base is within the arc region.
What matters for tactician is what path the shot takes. For tactician to trigger the shot must be, from closest point to closest, at range 2 and in arc.
You are taking an out of arc shot if the closest point to closest point, on a ship that can fire outside of its firing arc, is outside of arc. The shot follows the closest point to closest point (because that is what you use to determine if the shot is obstructed) and if that line is out of arc then tactician will not trigger because the shot was taken out of arc.