Here’s a rather simple way to look at it.
First, imagine that you are conducting a right hand visual approach to runway 36. Your track on the downwind leg would be 180(with speed and configuration appropriate) and the track on the base leg would be 270 (again, with speed and configuration appropriate). When the aircraft is about 45 degrees to the runway, you make the turn to base, etc, etc. Simple enough.
Now, you are flying the DME arc to this same airport. At some point during this ARC procedure, you will be “Downwind” (This point may be a lead radial / distance to the ARC). Then, at some point you will be on “Base” during the ARC procedure. Configure the aircraft appropriately and all works just fine.
The real issue is that of “Situational Awareness”. All too often people get caught up in the ARC itself and loose the basics of “Where are we” in relation to the bigger picture.
Keep in mind where you are, and the ARC becomes as simple as the visual approach referenced above.