A triple fusion takes three different personas. The game first sorts them by base level. The highest one waits; the lower two combine on the normal chart into an intermediate arcana. If those two share an arcana, the intermediate is simply that arcana.
The intermediate arcana and the waiting persona's arcana are then looked up on the triple chart, which differs from the normal one. The level target is the three base levels averaged, plus 4.25. You get the lowest persona of the final arcana at or above that target. If no persona is high enough, the fusion isn't possible.
When to use it
Triple fusion reaches higher levels than normal fusion from the same ingredients, thanks to the 4.25 bonus. It also opens arcana combinations the normal chart doesn't allow. Example: Pixie + Slime + Izanagi gives Fortuna (Fortune, level 35).