To note, Arcanist magic IS heavily dependant on geometrical shapes. That's been told to me in an Alchemist quest where you need to make a grimoire.
They really depend on special geometrical patterns drawn in enchanted paint, that is why they carry books full of them for more ease than requiring creation of those on the spot, also why grimoires are written with enchanted ink of various types, from silver-based ink to plant-based.

ALSO the usage of Egi... According to Fantasia it means 'To do, drive, lead', or to 'pass or spend time' in Latin... You know, saying "Ifrit Egi" would pretty much be the same as saying "Ifrit Driven" or "Ifrit Led", which is exactly where I think they went the "Powered by" sense of 'driven by'.

Of course i'm not sure if latin works that way but, it makes sense