I am not sure if this is just restating the obvious but hey...

If you have a triangle, you cannot move one of the corners relative to the other two without changing the length of a side, so it has to be rigid.

Looking at other shapes, in order to move one point the only way I can do it is by moving at least 3 others, in a triangle there are only two other points so you can't..

You could possibly proove this using cunning geometry but it is far too late, and I need to go to bed.