I tried to reproduce the issue, and the "tangent" point is (in my case at least) calculated from the endpoint of the original line. I thought the Perpendicular snap would do what the OP wants, but it doesn't register anywhere on the arc. Strangely enough, you can draw a line perpendicular to an arc.
I agree with Eldon, a temporary line should help to find the right point. Anything else adds complexity.