Sense 3 (“branch of a Chinese underground criminal society”) is due to the word being applied by the British authorities to underground society in Hong Kong based on the geometry of the Chinese character, derived from a name used by some of those societies, 三合會／三合会(sānhéhuì, “Three Harmonies Society”), referring to the union between heaven, earth, and humanity.
2000, David Pierce, Irish Writing in the Twentieth Century: A Reader, page 625:
There are, says the Irish triad, 'three fewnesses that are better than plenty: a fewness of fine words; a fewness of cows on grass; a fewness of good friends around good ale'. As an Ulsterman I would agree.
In his general deportment he was pompous and important, affecting a species of florid elocution, which often became ridiculous from his misarranging the triads and quaternions with which he loaded his sentences.
The point — and the symbolism — was that so long as the informer kept his mouth shut he was alive. Harry had seen the result of zjuk carried out by the Tapei Triad on a poor jerk they found in a back street of Tan-shui. They had used broad nail heads that didn’t make such big holes on their way in. When the paramedics came and pulled the brick off the dead man, the face came with it.