How do I join a helix to a line using an arc, with the line being perpendicular to the helix axis?

To do as you described, an arc tangent to both line and helix, the line cannot be on the same plane as arc and the helix endpoint. Create a 3DSketch, insert a line parallel to a plane perpendicular to the helix axis, insert a 3 point arc connecting the end of line to end of helix.  Then make tangent relations between the arc, helix, and line.  To get the arc endpoint coincident with the helix endpoint, you may need to use Convert Entity on the helix.  The correct solution will have a line offset down a little from the helix endpoint and the arc will be at an angle matching the helix angle.

See attached.

I'm not thinking too hard about this, so I may be missing something at this point in the day.

But, to achieve what I think you are wanting (a smooth transition between the line and the helix) I think you will need to insert a spline.  An arc is a segment of a circle, all the points of which will be in the same plane.

If you look closely at the attached part, I have inserted an arc, which is only tangent to the line, and a spline which is tangent to both the helix and the line. I also inserted a line segment tangent to the end of the helix. I added a reference plane defined by the line and the end of the helix (any arc tangent to the line and connecting it to the end of the helix will reside in that plane). You can see that the arc is in that plane and that the helix sweeps up out ot the plane.  So, the arc cannot be tangent to the helix and the line. The tangent line segment dives under the plane and highlights how geometry coincident to the plane can't be tangent to the helix.

If you want the transition between the helix and the line to be a true arc, you will need to vary the helix (like a spring with a flat end) so that the end is tangent to the plane which the arc will be in. That way the tangency between the arc and the helix only has to deal with planar geometry.

Erik,

I wasted so many hours on this problem. Then I had an epiphany as to why it wasn’t possible.

For an arc to fillet two lines, the two lies must be coplanar. If you have a line that itches a helix and goes off tangent to the helix‘s defining cylinder, then a tangent line for any further point on the helix will not be coplanar.