I have a 1984 trek 420 that currently has a double chainring. I want to install a Sakae CX triple. I'd also like to use a cartridge bottom bracket. The Vintage Trek sight says to use a 127.5mm axle that may require a 2mm spacer on the right bottom bracket end. That is with a Sugino AT crankset (triple). Does anybody know if it would be the same with the Sakae CX triple?

I'm not sure why you would need a spacer. The important thing is to observe the chain line, if it's too close to the frame, then I suppose the spacer is needed.
I converted an old Raleigh from a double to a triple and the only changes that I needed to make were (aside from the crank/chainring, front and rear derailleurs) was I swapped out the 113mm bb assembly (which was loose ball bearings style to the longer axle sealed cartridge version 122.5...IIRC

Another symptom is chainrub on the middle chainring when on the small. I did a couple conversions and pushing the crank farther out helped.

I did a triple conversion on an early 80s Trek 700 with a early shimano 600 double with a triplizer and found the 127.7 was too wide and a 124 worked better.

My experience with triples is to have a variety of spindles to work on the fit. I picked them up at various swap meets, and some are bad, but thats ok for fitting up the triple.

Many of those older cranksets used asymmetric bottom brackets. So a spacer allows a modern cartridge BB to mimic the asymmetric.

The empirical approach described above is a good one. Find a used cartridge that you're pretty sure will be too long. Try the local bike co-of if there's one near you. You're not going to ride with it, so it doesn't matter if the bearings are bad, provided that the threads on the cups are in good shape. Install the "test cartridge" in the BB shell and bolt the crankset to it. See how much clearance you have between the inner ring and the chainstay. You want at least 2mm of clearance, but probably no more than 4mm.

As an example, say you install a 127mm cartridge and have 6mm of clearance. (You can use an allen wrench as a feeler gauge for this.) In that case, you want a cartridge that's about 3mm shorter on the drive side. That means it will also be 3mm shorter on the non-drive side, or 6 mm shorter overall. Remove the test cartridge and replace it with a 120-122mm one that you've purchased new.

Don't worry about precision too much--there's a fair amount of leeway here. When in doubt, it's better to go a little short than a little long. If the inner ring is too close to the chainstay, you can always increase the clearance by adding a 1 or 2mm spacer under the drive side cup. But if it's too long you have to live with it or get a shorter spindle.

To check whether a given BB combination will work, I agree you have to look at clearance between the inner chainring and the chainstay, but you also have to look at the chainline, i.e. the distance from the center of the downtube or seat tube and the center of the middle chainring.

Thank you everybody. I do have the axle that came with the triple crank. I'll try that first to get an idea of the length I need. Should the distance from the downtube center to the center of the middle chain ring line up with the middle of the rear derailleur?

Right, the correct chainline would be something like 43 mm between the center of the middle ring and the center of the seat tube. But clearance is easier to measure, and in my experience getting the clearance right also means that the chainline will be right. Shamelessly empirical, as I said, but there's no need for a high degree of precision with this.

If you have the original BB axle from your triple, measure it end to end and obtain a UN-26 or other BB cartridge of the approximately the same length. Better to be 1-3 millimetrs wide than too short.

Good luck.

The triple came off a Schwinn La tour Luxe and it's going on a Trek 420. Not sure if the axle is the same width as what the Trek would have had with a triple.

Per Sheldonbrown.com

"The spindle length mainly depends on what kind of crankset you'll be using, and doesn't have much to do with the frame. "

https://www.sheldonbrown.com/bbsize.html

Thanks for that info oddjob2. I'll try that axle first . If it fits I might just use that instead of a cartridge.