Triangular formations for modern armies have been in place since the time of Napoleon. They work on the basic principal of fire and movement. In theory you have a three to one advantage over your opponent when on the attack, whatever the size of the engagement. Moreover a triangular formation is an inherently flexible organization. It allows one element to make the assault, one element to provide fire support, and one element to be held ready for exploitation. It works in the defence or when retrreating.
If we wind back the clock a little to musket and ball era, the concept of triangular formations can be traced back to the three ranks of the Infantry formation. Whether falling back, or advancing, the front rank is firing, the middle rank is sighting up, and the rear rank is reloading....this is where the concept comes from.
Without a doubt, a triangular organization is a far more efficient formation, on one proviso. It assumes some degree of manouvre is available. During the advance on the marne in 1914, this was certainly the case, and the efficient triangular divisional organizations in the french army at that time is probably what saved it in that first offensive. However during WWI generally, the triangular TOE theories did not work complatrely as planned, because for most of the conflict there was very little manouvre, and a lot of close in hand to hand fighting. There was also a lot of casualties suffered so ready access to larger battalions, with large reserves of men, and a large fpf (final protective fire) advantage was found more important than inherent flexibility and mobility. This is why the US Army and many allied armies retained a quasi square formation at both divisional and corps level. Firepower and reserve were more important for most of the war than mobility. Whilst the turkish army had indeed resolved to reorganize its army in 1910, this was far from complete AFAIK, and in fact the turks fought Gallipoli and most of their campaigns in Palestine, anatolia and Iraq with a fairly immobile square formation TOE. In th relatively mobile operations of Palestine, this definately worked against the turks....time and again they found they could not react nearly quickly enough to manouvre operations initiated by the allies. Dont know what happened on the Anatolian front, in Gallipoli if the Turks did use a triangular structure it certainly is not mentioned in any of the accounts that i know of (but square formations are mentioned), and it makes sense that for a front like Gallipoli a TOE that favours firepower would be of greater advantage over a TOE tailored for manouvre. The germans might have been trying to give the turks bad advice, but the turks knew better (thats a bit tongue in cheek guys....Sanders was a definite asset to the turks in that campaign....)
A link for some reading up on the theory and history perhaps
Sixty Years of Reorganizing for ... - Google Books