Cleaning the Shower Door- a Series of Experiments

It's been a while since I've cleaned the shower door, as you can see in this picture:

The shower door is fairly new, and I still haven't landed on a completely satisfactory cleaning approach.  I reasoned that the cloudiness was caused predominantly by soap scum, so I stared the cleaning process with soap scum cleaner and a cloth rag.  It got some of the cloudiness, but it still wasn't clean:

Next I went after it with Windex and a paper towel.  Again, some improvement, but not great:

Then, I decided to try a trick I read about on the World Wide Web.  One person who had worked in a motel described how they always cleaned their shower doors using a dryer sheet that they'd moistened with water.  I decided to give this a try.  I found that the dryer sheet seemed to loosen more of the cloudiness, but I needed to follow up after it with Windex again to remove the loosened gunk.  Here is the result:

A nice shiny door.  My recommendation: go straight for the dryer sheet followed up by a swipe of Windex.