Lloyd Beachy sends us a LISP routine that he uses to create arcs that are tangent to existing lines.

"There are a number of tools I use to draw over the top of images inside AutoCAD. I have a similar tool that is fun to use and can save just as much time: Dynamic Tangent Arc. Select an existing line or arc and a new arc will be drawn tangent to the nearest endpoint. The dynamic part of the function is where you can adjust the tangency point by pressing the + or - keys on your keypad. This will either lengthen or shorten the existing entity, shifting the tangency point accordingly (represented by the dashed purple line). The amount of adjustment is based on the zoom scale; if you need finer adjustment, simply zoom in a little further."
 
Notes from Cadalyst Tip Patrol: To start this tool, load the routine DTANARC.LSP with the APPLOAD command. Once loaded, type DTA to start it. Select an existing line or arc, then pick the endpoint of the new arc. Before you pick the endpoint of the new arc, you can adjust the point of tangency by pressing either the + or the - key to lengthen or shorten the existing line or arc.