the one where you gotta defend the keystone from attacks from 3 sides?

trying to mass dragoons with a few arbiters doesn't seem to work

I'm not sure what difficulty you're playing on but this strategy worked on brutal:

Expand asap and get the chrono surge ability to saturate your two bases very quickly. Cannon/shield battery defend the right side and mass up dark templar (the ones that teleport around dealing a billion damage) and corsair. Lightly defend the other two entrances while also using your army to defend these. They had very little detection when I did it and the dark templars just shredded everything and the corsair disruption web (which they auto-cast) stops them from attacking anyway. Since they can't see the dark templar, they just get pushed back and don't actually attack the ramp so the AI defenders can actually amass pretty decent stuff. And corsair just shred everything in the air. Eventually you can get lots of defense for each nexus. I didn't lose any nexuses using this strategy.

hadn't thought to use corsairs, will try this, thanks!

make dark archons/tempest mind control their colossus/tempest/carriers/guardian
excess minerals goes to cannon to fight off ultras

Mass vanguard corsair done that shit on brutal with like zero units lost