Was once told by a history teacher that a certain type of bayonet was banned by the geneva convention, but there were certain countries in Asia who were not signatories of certain chapters and therefore continued to use these (triangluar?) bayonets.. Is there any truth in this?
So there are several issues here. The first is that this isn't something that would fall under the Geneva Conventions, which are actually the "Laws of War" in relation to non-combatants, that is to say the wounded, the captured, and civilians caught in the warzone. The Hague Conventions are the major "laws" which regulated warfare through the 20th century, and include prohibitions on various things, such as dum-dum bullets. But as far as I'm aware, and after a quick confirmation of the text of the 1899 and 1907 Conventions, bayonets are not dealt with in either text. Even that aside, the major Asian powers, such as China and Japan, were signatories.
I did some poking around to see if this was a common belief, and it does seem to be propagated on the internet with regard to triangular bayonets, but without any basis. I would speculate that it is possibly a result of some 'historic telephone' regarding the debate over the saw-tooth bayonet which occured during World War I. I wrote of that here, but to keep it very short, the French didn't like the Germans using them, so attempted to interpret a clause of the Hague Convention which banned weapons designed "To kill or wound treacherously individuals belonging to the hostile nation or army" as applying here. It was not resolved though, and the bayonets not declared illegal, although nevertheless phased out of German use. The same passage could also be construed against triangular bayonets, but certainly isn't explicit.
I'd like to expand what /u/Georgy_K_Zhukov just posted. First, he is right in asserting that bayonets were not typically covered under the conventions, and indeed, you can read them online in their entirety.
Triangular bayonets were never intended to be more deadly than any other kind, it was simply that a wedge of steel was easy to produce and nearly indestructible, which was attractive to the men whose job it was to pay for and equip soldiers. It was this expedient more than any others that accounted for the ubiquity of wedge-style bayonets.
Even given their ubiquity, bayonets accounted for a very small margin of battlefield casualties, something around 1% of those killed and wounded. This happened during the brief clashes that characterized bayonet charges(which I've posted about here and here), because men were trapped in a place they couldn't easily flee from. Almost all of the men killed by bayonets at Lexington and Concord, for instance, were bayonetted when the British flank parties were able to catch Americans unaware inside homes they were firing from - they couldn't simply run away, and so they were spitted.
The reason that armies started favoring blade-style bayonets was due to another logistical concern: the soldier's load. If the bayonet can be made into something more widely useful than stabbing a guy, it was easier on the army's pocketbook and on the soldier's back. It led to some strange innovations, such as the "trowel bayonet" made for the US army. More usually, a blade style could double as a knife, a nice all-purpose tool that can be used in far more ways than a wedge of steel.
Casualty figures comes from Redcoat: The British Soldier in the Age of Horse and Musket by Richard Holmes.