Seeking software to help triangulate segment DNA matches?

The closest to what you are asking might be GEDmatch's new "Find common ancestors (MRCA) from DNA matches" tool, which they added to their Tier 1 toolset on July 1, 2020.

What this tool does is compare all the people in your tree that you've uploaded using GEDCOM to GEDmatch, with the trees of all your DNA matches. It seems about 10% of GEDmatch users have added trees to their accounts.

The results include excellent information about each of your DNA matches who have a match in their tree to you.

Unfortunately for me, I have no matches, but from what I hear, I am the exception rather than the rule.

There is a Family History Fanatics tutorial video for it that includes examples where there are matches.

Your criteria for comparing the tree of three people for common matches is very limiting. Unless two of those people are closely related, it is unlikely you'll find three people matching. The purpose of DNA matches is to find potential cousins who share part of your tree with you. So comparing your tree with the tree of one match at a time will accomplish that goal.

Also, the work you are doing to enter your match's complete tree into your family database seems quite excessive to me. You don't state where you are getting their trees from, but if you simply searched through their name index and/or place index, you should be able to identify potential common ancestors without building their entire tree.

The GEDmatch "Find common ancestors (MRCA) from DNA matches" tool is not free. You must subscribe to their Tier 1 services to use it. But you can subscribe for just a month when you need to use this tool.

I don't know if there's anything out there yet that is quite as robust as you're suggesting, although I'm also hoping to have something like this in the near future.

For now, I'm using GenomeMate Pro.

https://www.getgmp.com/

It imports data from all of the major sites... FamilyTreeDNA, Ancestry, 23andme, MyHeritage, and also GEDmatch & DNAGedcom.

It's not so much a genealogy database like Legacy or Family Tree Maker. It deals more with the DNA. It does allow you to show overlapping segments & triangulation groups. You can enter surname information so that you can see which surnames each of these groups share.

There is also the capacity to upload GEDCOMs for your matches so that the information is easily accessible.

You may upload all data you have to GENi.com or MyHeritage.com SaaS. They have unique and very powerful feature as finding of possible duplicates of persons. The question is that both services are limited in downloading of DNA, but it is possible too. Another option is to use GEDMATCH service and it's triangulation features, but they are no very user-friendly or convenient.

There is a very useful plugin for Family Historian called 'Find Duplicate Individuals' that does the majority of what you want. It is not intended for genetic genealogy but is a general tool that I use frequently in my surname study which does unintentionally create duplicates.

It creates a result set that looks like

You can then run the merge individuals function to compare the fact data between the two individuals and that result looks like this

Not exactly what you want but pretty close. Unfortunately the free trial of Family Historian does not allow the running of plugins so you would need to purchase it to try it out to see if it meets your needs.