How do I round one corner of a triangle? Dragging 1 widget rounds all corners and holding down Alt/Option doesn't help.
In the comments you mentioned that it was a "Illustrator trial version activated a few hours ago", which would be Illustrator CC 2017.1. In this version what you were doing (dragging one widget) was the proper way, but you need to select only one of them instead of all of them. You can do this with the black arrow (V) by selecting the object and holding CTRL/CM. This will make the "widgets" visible, as circles. While still holding CTRL, if you click on the widget for the corner you want rounded it will turn solid. This means it is selected and can be individually manipulated.
make a fresh triangle (you can do it with the polygon shape tool, click art board and enter 3 sides)
Make sure it is unselected. Drag a box around 1 point with the white arrow.
Now convert anchor to round and adjust the handles (the hard way) or on my new version of illustrator CC when I select a corner with the direct selection tool (white arrow) it shows a "Corners" tool in my top bar. I can increase and decrease the amount of roundedness on 1 tip of my triangle with it.
Here's the hard worker's solution. Nothing clever, but no need to tinker with nearly invisible - that's the Bezier handles in a sharp corner.
Addendum for a non-constructive comment: Having a 2010 version of Ai and no interest for programming (did it for work as an Assembly coder) I show in one image, what it takes to position the wanted rounding circle (=the red one) tangentially by relying on Ai's smart quide snapping. Anybody who is familiar with elementary plane geometry sees immediately the idea: