While NumPy is of course not normative, this is what they do:

>>> numpy.random.triangular(left=1, right=2, mode=0)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "mtrand.pyx", line 3218, in mtrand.RandomState.triangular (numpy/random/mtrand/mtrand.c:13407)
ValueError: left > mode
>>> numpy.random.triangular(left=1, right=2, mode=3)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "mtrand.pyx", line 3220, in mtrand.RandomState.triangular (numpy/random/mtrand/mtrand.c:13433)
ValueError: mode > right
>>> numpy.random.triangular(left=1, right=1, mode=1)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "mtrand.pyx", line 3222, in mtrand.RandomState.triangular (numpy/random/mtrand/mtrand.c:13459)
ValueError: left == right