You could probably use snapping and align the working pivot to the edge - however I use FunkyBunnies' FB_SubObj script (from a Quad menu), it's insanely useful.
Max does "sort of" have a native function for this, it's just really, really bad. You'd use a "Working Pivot" to first define the pivot, and then you make your edits. Really not ideal, which is why you'll find so many scripts that do it a better way.
It's also utterly useless on flat geo. So the example of the triangle, and aligning the pivot to one of the angled edges can't be done this way.
I can think of very convoluted ways to achieve this natively, but honestly... it's not worth it. People like Chris Whitaker (FunkyBunnies3d) have already done the hard work.
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I tried that, and it wasn`t running for me.
http://www.garydave.com/jing/Working_Pivot_002.swf
It's also utterly useless on flat geo. So the example of the triangle, and aligning the pivot to one of the angled edges can't be done this way.
I can think of very convoluted ways to achieve this natively, but honestly... it's not worth it. People like Chris Whitaker (FunkyBunnies3d) have already done the hard work.
http://www.scriptspot.com/3ds-max/scripts/align-pivot-to-direction
http://miauumaxscript.blogspot.jp/