A post based on my responses at the Autodesk Forum: Tangent Circle to Tangent Circle.
It could be easier...
I see Revit behaving this way, they regard the first point as ineligible to being tangent because it depends on the bearing of the line, With that assumption or bias, the first point is necessary to make a tangent condition possible. I can easily snap to a location on the circle (a pulley for example) that couldn't be tangent to the next pulley.
AutoCAD deals with this in a clever fashion (when we invoke the tangent snap) by fixing (changing) the first point to be tangent after the second point is placed. If we aren't careful with our second pick point (snap tangent too) the tangent line might end up on the opposite side of the pulley.
In contrast, Revit handles it naively, because it regards our first point as ineligible to tangents because it isn't considering this particular end result: "I want to draw a line tangent to two circles". AutoCAD appears to know this by virtue of snapping tangent for the first point so it can adjust the final bearing, and attachment to the circle, of the line.
To get around this naiveté, I place the first point on the pulley where it looks like it can be tangent, to my eye. The second point snaps to tangent with the icon. I return to the first point and grip/drag it away and back to let the snap icon appear, to fix it for tangent, just to see if I was close. If my guess wasn't accurate, it is now.
After reading a reply to my comments I did a quick sketch in AutoCAD and then did the same sketch in Revit using the same pulley sizes and offset from one another (see Footnote). The tangent lines have the same x/y properties for start and end as the AutoCAD version, that I made using its snap tangent.
This is the native DWG sketch and properties screen captures for each element.
This is same information but for the Revit drafting view exported to DWG. When I create an External Reference of the exported Revit drafting view it lands right on top of the native sketch. If you look really closely you'll see a value is slightly different in the Revit version. I think that might be my fault, sketching. Regardless, I think close enough is fair.
Footnote: Regarding a drafting view aligning with a DWG file after export: It might not be obvious but drafting views have an origin. To test that claim link a DWG, that has a marker at the WCS origin, into one and you'll see where the origin is. I did that before I did any sketching so I'd know how to place the pulleys in the same place. That made it possible to compare the tangent lines after exporting it to DWG.
Also the Start and End X/Y values are reversed. That's either just how Revit interprets the vector of each line segment or it's because of the direction I chose to sketch them in Revit. In AutoCAD I started at the smaller pulley. I didn't make sure to sketch in the same way in Revit, sloppy scientist.