This is a tricky problem, no easy solution with any package, not even Illustrator.

I want to be able to find the tangent to a circle at a particular location, then place a second circle at the same tangent at the exact same spot. Easy to do if the tangent is vertical or horizontal, much harder if the tangent is at any other angle!

I can guestimate the location, rotate so that the rough tangent is horizontal, locate precisely, and then rotate back, which is very very slow. Alternatively, I can blow up to 51200% and locate at a level of accuracy in which the error is too small to worry about (also slow, especially as the Intaglio display is temperamental at high magnification levels).

Can anyone come up with any good tricks that can get a good solution with less work?

Tick ‘Snap to Points’ and Show Grid (Layout menu).

Choose the Direct Selection tool and select the 4 points on the first circle.

Drag the first circle so that the tangent point is at the intersection of 2 grid lines.

Select the points on the second circle and drag until the tangent points align. When the 2 tangent points are circled red (see screengrab), they are aligned.

Thanks for the suggestion. I can adapt this to get it to what I want. One circle can’t move, and the other will almost always need to be a different size. So, I need to rotate the first circle so that a control point is close to where I want it to be, then rotate the other one to the same amount, then place them.

But, I also need them to be arcs so that I can reshape them, so I need to work with multiple copies, one circle, one converted to stroke.

Hope that was clear, but this was the result I was working for:

Tick ‘Snap to Points’ and Show Grid (Layout menu).

Choose the Direct Selection tool and select the 4 points on the first circle.

Drag the first circle so that the tangent point is at the intersection of 2 grid lines.

Select the points on the second circle and drag until the tangent points align. When the 2 tangent points are circled red (see screengrab), they are aligned.

I probably don’t understand your method—it seems very complicated! As I understand it, you’re creating curved paths from circles.

First of all, draw a circle and convert arcs to curves. Zoom in and drag one circle until it just overlaps the other.
Unite the circles and delete the unwanted points. This way, Intaglio inserts tangent points where the circles meet. You can position one circle anywhere relative to the other—there’s no need to rotate until the tangent points align. Circles can also be scaled until the desired result is obtained.

The 2 points between the remains of the circles need to be joined. I wasn’t able to create just a single tangent point.

When you outline the path, you might find that extra points appear.

I tried Illustrator and Freehand and got more or less the same results.

Firstly, widen the stroke. Then go to Object>Convert>Stroke to Fill. The linked screengrab goes into more detail. You might find this easier to do if the palettes are on the desktop.