There are tricks to all trades, they say, and I think I finally got lucky and found the one that let me get Linux Mint 17.2 "Cinnamon" to recognize Windows 10 and install alongside it. And I'm so proud of myself right now, it would just curdle your cream! I've only been about a week trying to outsmart the system and get it to do what it says it can but didn't, until now.
I found the answer in a post on the web on a page called "Easy Linux Tips Project" and I probably owe Microsoft an apology for some of the nasty things I've said here lately. Not because they never make mistakes, but because they sure as hell aren't the only ones, as this past week would prove. If I wasn't such a bull-headed old fart, I'd never have persisted long enough to stumble over the answer to the problem. It was under a heading that went something like "How to install Linux Mint in a dual boot with Windows when it seems impossible."
The secret lies in the live disc of the install DVD for Linux Mint. On that live Demo that you get to play with while deciding whether or not you want to try actually installing it, there's a working version of GParted, their own Partition Tool, and when all else failed, I tried running that, just to see if it could recognize the two Windows partitions already in use on here. I was very pleased to see that it did see them, and accurately described their sizes and formatting. It also saw the "unallotted" other partition I'd left for the Linux install. So I let it sit like that for a few minutes, just to get it used to the idea that there really is other stuff on the drive.
Then, when I switched back to the "Install Linux" icon, and it showed me the windows for it, this time it finally said it had found Windows and would I like to install Linux alongside it. Something it hadn't been showing me before. So then I naturally said, "Yes, you tricky little (censored!), Go for it!" And now, there's both Windows and Linux on here, each with its own boot loader, and each working very nicely. And I win! It can be done after all....
"And why did you want to?" you ask. Because.....it wasn't easy, and because this computer has a 64-bit quad core processor that can probably run both of these 64-bit systems better than my other dual-core PC. I'll know very soon!