Hi Steve, you'll want to have Straight Snap enabled to draw a line tangent from a point on a curve.

With it enabled (highlighted in orange in the bottom toolbar), you can then get a tangent tracking line from that point. If you don't get one, then check to see if you have possibly disabled tangent snapping under Options > Snaps > Straight snap options.

Here's a demo of how it works:




If you want to make a tangent line that passes through a specific point, then in that case you start at the point and use Object Snap to find the tangent point on the curve, like this:




Then the other type of tangent snapping is that double tangent one that you saw - that's when both the start point and the end point are on curves, and there is a line of shared tangency somewhere in the general vicinity of both points. This snap not only snaps the end point of the line but also adjusts the start point of the line as well. Usually the target point that triggers it is found slightly to one side of the single tangent snap like this:



- Michael