Why would I approach a random muscular guy? What if he’s dangerous? I don’t know anything about him. He might as well be a horrible person that could endanger me. No thanks.
Not sure where you gleaned an obsession. I haven’t thought about him for over a month since I barely saw him at the gym. If he’d carried on mostly ignoring me and not acting unusual, I wouldn’t be thinking about him again at all. But since I don’t have any other attractive men my age around, why not focus on the one guy who’s around sometimes? Women need some entertainment, too, and at the very least this situation offers some novelty.
I've been the guy on the other side of this. On a group bicycle ride, I spent some time talking with a woman. At the end of the ride I suggested we get together again. She had a kind of hesitant expression but said that she didn't know me well enough.
Like that cyclist, you don't want to approach the guy because you don't know anything about him, but it's clear that you won't know anything about him unless you approach him. You're hoping to find true love -- with a co-worker or a friend of a friend, a very limited pool.
Obviously, women need to be more concerned about psychos than we do. You should follow the standard advice of meeting him in a public place and not putting yourself in a dangerous situation. But beginning to talk with him in the gym is safe.
I agree with Skater's advice that you should approach him. One possibility would be to follow up on the unusual situation that he was there with someone: "I remember seeing you here several weeks ago, showing your brother how to use the machines, but I don't think I've seen him since. Did he pick some other fitness activity, or did he just decide that he couldn't keep up with you?" Maybe he'll say "My wife was leaning on him to work out but he didn't like it" and there's your answer. More likely, you two have a casual conversation. Then the next time you're both at the gym, it's easier for either of you to smile, say hello, and start talking again.
You have an advantage over the cyclist woman I mentioned: You have good odds of seeing this guy again. If he seems interested in talking with you, then over several gym visits you can get past the "don't know anything about him" stage. Then there might be coffee or a dinner in a public place. And if nothing gels (e.g., he's not attracted to you), then you just go back to enjoying the eye candy.
