Hi Moi3dFan, both single Tan and also Tan/Tan snapping work on 2 points like when you are drawing a line.

The single Tan snaps the 2nd point of the line onto a curve where the curve tangent passes through the first point.

Tan/Tan snapping snaps both the 2nd point of the line and also repositions the 1st point of the line onto curves where there is a line of shared tangency between the 2 curves.

Both of these affect 2 picked points, they don't involve the geometry of the object you are dragging.

You can get the result you want though by using a perp/perp construction line, see these video examples:

http://moi3d.com/forum/index.php?webtag=MOI&msg=11056.24
http://moi3d.com/forum/index.php?webtag=MOI&msg=11056.25

And another method - first place the object at the center of the circle and then you can use Transform > Move with a perp snap onto the circle like this:
http://moi3d.com/forum/index.php?webtag=MOI&msg=11056.26

- Michael